Assuming you didn't go insane, your sense of time would adjust. Remember, there's no clocks or calendars either.
Let's flip the scenario around. Say you had been given immortality today. Right now. You're still going to end up living for a million years. What if there's a nuclear war tomorrow? Arguably a worse situation that being 1-million years in the past.
You want immortality? You need to accept that things are going to be utterly and completely fucked at some point. And there are going to be long stretches of time where you're completely and utterly alone. That's just the nature of it.