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The Theurgy Wizard from UA: Wizard Revisited gets to choose a Cleric Domain at 2nd level. At 14th level they get the Arcane High Priest feature, which grants them the 17th-level benefits associated with their chosen domain.

A cleric who chose Arcana as their domain would therefore gain the Arcane Mastery feature, which reads as follows:

ARCANE MASTERY

At 17th level, you choose four spells from the wizard spell list, one
from each of the following levels: 6th, 7th, 8th, and 9th. You add
them to your list of domain spells. Like your other domain spells,
they are always prepared and count as cleric spells for you.

I've seen people on the internet argue that this 14th level wizard would be able to cast the chosen 8th and 9th level spells. This is of course not true, as the wizard in question would not have the necessary spell slots, regardless of whether those spells are always prepared.

But this wizard could pick up three levels in another Full Caster class (bard, cleric, druid, or sorcerer) or six levels in a Half Caster class (paladin, ranger, or this new UA revised artificer). Per the Multiclassing Rules, this would give this character a 9th level spell slot.

Question:
Can this hypothetical character cast their 8th and 9th level spells granted by the Arcane Mastery feature via the Arcane High Priest feature, despite not being able to know or prepare any other spells of that level?

Disclaimer: I know that UAs are playtest material and not tuned for multiclassing. I know that the situation I describe in this question is very unlikely to come up in a real-life D&D game. I'm interested in an answer per RAW. I really like designing (sub)classes, (sub)races, feats, magic items, etc. for my game. So I need to know how the D&D 5e rules behave under extreme circumstances, hence this convoluted question.

It's complicated but maybe you can cast these four spells if you multiclass

You've already noted the Theurgy Wizard feature that is granting you the 17th level cleric domain feature. So, if you picked the Arcana Domain, you would gain the ability you quoted (I will add emphasis)

At 17th level, you choose four spells from the wizard spell list, one from each of the following levels: 6th, 7th, 8th, and 9th. You add them to your list of domain spells. Like your other domain spells, they are always prepared and count as cleric spells for you.

So what does this mean? You add these four spells to your cleric domain spells. The issue is, you never got the Divine Domain cleric class feature or, in particular, the Domain Spells feature. The only interaction with Domain Spells that the Theurgy wizard has is from the Arcane Initiate feature:

(...) whenever you gain a wizard level, you can replace one of the wizard spells you add to your spellbook with a cleric domain spell for your chosen domain. The spell must be of a level for which you have spell slots. (...)

Given this, there are two ways to interpret what the Arcane Mastery is giving a Theurgy Wizard:

1. 
   

   The first idea is that looking at the Domain Spells feature for
   a cleric...

   
   
   

   
   

   Once you gain a domain spell, you always have it prepared, and it doesn’t count against the number of spells you can prepare each day.

   
   

   
   
   

   ...the Arcane Mastery sentence "Like your other domain spells, they are always prepared and count as cleric spells for you." is just a reminder of what being a domain spell means for a cleric. So, like other domain spells, since the Theurgy wizard does not actually have the Domain Spells feature, a Theurgy Wizard does not have these 4 spells always prepared and this simply adds these 4 spells to the possible choices one can make with Arcane Initiate (effectively making the Arcane Mastery ability mostly useless for a single classed Theurgy wizard).

   
   
   

   Still following this interpretation, if you choose to multiclass into cleric, now you do have a Domain Spells feature so you should also have the 4 chosen spells prepared as a cleric, allowing you to cast them once you have the available multiclass slots. Otherwise,if your multiclassing does not include cleric, you would not have these 4 spells always prepared.

   
   


2. 
   

   The second line of thought is that this is a more specific ability that should work regardless of the lack of the Domain Spells feature. You could also consider that without a Domain Spells feature, Arcane Mastery is adding these spells to an empty list.¹ Regardless, you become a 14th level wizard that always has these four spells prepared, and they count as cleric spells for you.

   
   
   

   This means that because you are multiclassing to get spell slots of a high enough level, even if you have no cleric levels, you should still have these cleric spells always prepared and available to cast.

   
   

Conclusion and comments

I would recommend the first interpretation I presented because it creates less confusion and it seems more natural to me. In my opinion, unless I missed something, there really isn't much more you can go to in terms of what the rules are saying and it's going to be up to how people interpret what is written.

Since you state that you like homebrewing, I will suggest what I see as the root cause of this mess: Theurgy allows a wizard to obtain arbitrary (meaning from any cleric domain) features from a subclass for a different class.

When you just give a class features from a subclass from another class, you might run into issues when those features depend on other class features (that you might not have remembered to provide). This was probably not well thought out when creating the Theurgy Wizard. Consider also the Order Domain on a Theurgy Wizard. The 17th level feature requires you to deal Divine Strike damage which the wizard never gets access to, making it unusable.

In short, don't just give features from a subclass to a different class (or be extremely careful and selective when doing so) because they might not make much sense otherwise, as is the case here.

^{1. Suggested by John Clifford in a comment}

Yes, you can do this by multiclassing into another caster class

Let's look at this in sequence.

As a Theurgy Wizard, you hit 14th level and gain Arcane High Priest, which gives you the Arcana domain's 17th level feature, Arcane Mastery. This allows you to add a 6th, 7th, 8th and 9th level spell to your list of domain spells, so they will always be prepared.

Domain Spells

Each domain has a list of spells—its domain spells—
that you gain at the cleric levels noted in the domain
description. Once you gain a domain spell, you always
have it prepared, and it doesn’t count against the
number of spells you can prepare each day.
If you have a domain spell that doesn’t appear on the
cleric spell list, the spell is nonetheless a cleric spell for you.

PHB p.58

The section on spell slots in chapter 10 states the following:

Regardless of how many spells a caster knows or
prepares, he or she can cast only a limited number of
spells before resting. Manipulating the fabric of magic
and channeling its energy into even a simple spell is
physically and mentally taxing, and higher-level spells
are even more so. Thus, each spellcasting class’s
description (except that of the warlock) includes a table
showing how many spell slots of each spell level a
character can use at each character level. For example,
the 3rd-level wizard Umara has four 1st-level spell slots
and two 2nd-level slots.
When a character casts a spell, he or she expends
a slot of that spell’s level or higher, effectively “filling”
a slot with the spell. You can think of a spell slot as a
groove of a certain size—small for a 1st-level slot, larger
for a spell of higher level. A 1st-level spell fits into a slot
of any size, but a 9th-level spell fits only in a 9th-level
slot. So when Umara casts magic missile, a 1st-level
spell, she spends one of her four 1st-level slots and has
three remaining.

PHB, p.201

So you have those spells prepared, but you don't have the spell slots to use them: at 14th level you have a 6th and 7th level slot, but no 8th or 9th. You would, as you said, have to gain 3 further levels in a caster class to get those slots (whether that's going to 17th level Wizard or multiclassing 3 levels in a different one)

Assuming you do take those three multiclass levels and then get to 14th level Theurgy Wizard, you would have the spell slots to cast your prepared spells and so would be able to use them at that level.

The potential argument against this comes from the section on multiclassing spell slots:

If you have more than one spellcasting class, this table
might give you spell slots of a level that is higher than
the spells you know or can prepare. You can use those
slots, but only to cast your lower-level spells. If a lower level spell that you cast, like burning hands, has an
enhanced effect when cast using a higher-level slot, you
can use the enhanced effect, even though you don’t have
any spells of that higher level.

However, in this case it's given you spell slots of a level that's appropriate for spells you can prepare, because you always have them prepared. The rule that Cleric domain spells are always in your prepared spells list circumvents the rule that you can only choose to prepare a spell if you have a slot of that spell's level or higher. You're not choosing to prepare it.