"Immortality" is a strange concept to talk about when the afterlife is known to exist. (Remember, Asterion isn't "alive", and hasn't been for thousands of years!)
I think that immortality in the sense of *living* (in the real world) forever would be both unlikely, and problematic for a lot of reasons. (Remember Asterion's backstory, and what happened when someone *else* wished for immortality....) But given that there are many afterlives, it seems reasonable to hope that the MC and Asterion end up in the same (pleasant) one.
It's plausible that this could be Hades--it's where Asterion should rightfully be. Logically, since the MC has firsthand knowledge that the Olympian gods are real, has offered a sacrifice to them (at Argos's behest), and might plausibly take future actions that could be regarded as worship of at least *some* of them (e.g. actively participating with Asterion in a pre-meal offering to Hestia), it's entirely possible that the MC could end up there as well. They could either go to the Asphodel Meadows, or the gods might decide that both the MC and Asterion took such heroic actions with the hotel to warrant Elysium.
An alternative is that the Labyrinth itself is, in a very literal sense, Asterion's personal hell. But we're not entirely sure what the power of the gods *with* the ichor is, and the extent to which *they* can break or creatively bend the contracts. Maybe a majority of the Olympian gods could rewrite the laws of the realm to make it pleasant instead, and the MC and Asterion keep running the hotel forever.