Dynamic pricing, so it's somewhat hard to get pricing ahead of time. Uber's estimate calculator on their website should give you a decent ballpark.

At 1am, there still won't be any issue getting an Uber or a Lyft at that hour. You can typically request as you arrive into the main terminal from where you land. Use any one of the referral codes for a discount although it's not much (I think around $5). Haymarket is far enough that even the filtering drivers shouldn't give you a hassle about it. It's honestly really straight forward - just make sure you match the plate number with the car you get into.