Bsrgvty

Fabled Passage has this text:

[T], Sacrifice Fabled Passage: Search your library for a basic land card, put it onto the battlefield tapped, then shuffle your library. Then if you control four or more lands, untap that land.

Amulet of Vigor says:

Whenever a permanent enters the battlefield tapped and under your control, untap it.

With Amulet of Vigor + 3 lands on the battlefield, can I play Fabled Passage, sacrifice it, find a basic land which enters tapped, Amulet triggers and untaps the land, tap it for mana (floating one), and then untap the land again using Fabled Passage's effect?

Yes, you can get 2 activations out of the land you fetch with Fabled Passage if you also control Amulet of Vigor, though technically a little different than you describe.

When the land fetched with Fabled Passage enters the battlefield, it always enters tapped. That means Amulet of Vigor triggers, but the Amulet's ability does not go on the stack yet. It waits until Fabled Passage's ability has finished resolving:

116.2a Triggered abilities can trigger at any time, including while a spell is being cast, an ability is being activated, or a spell or ability is resolving. (See rule 603, “Handling Triggered Abilities.”) However, nothing actually happens at the time an ability triggers. Each time a player would receive priority, each ability that has triggered but hasn’t yet been put on the stack is put on the stack. See rule 116.5.

Fabled Passage finishes resolving, and the land you fetched is untapped when it has finished resolving. Since an ability resolved, you as the active player get priority:

116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

Before you get priority, Amulet of Vigor goes on the stack. You can now tap the land for mana and float it, then let Amulet of Vigor resolve, untapping the land again. You can now tap the land again for the second time. and use the mana as you see fit.

No, not that way, but the other way around it works. You can't do anything else during the resolution of Fabled Passage's ability:

608.2c The controller of the spell or ability follows its instructions in the order written.

but since the land enters the battlefield tapped, Amulet of Vigor's ability will trigger and be put on the stack after Fabled Passage's ability resolves.

603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.”

You can tap the land for mana after Fabled Passage's ability is fully resolved but while Amulet of Vigor's ability is still on the stack, floating one mana and have the Amulet untap the land so you can use it again the same turn, effectively generating two mana.