It's a rental.

With that in mind, my go to choice would be an undermount. Second would be a drop-in. Never ever would I put a vessel in a rental. Do consider that if the vanity was designed for a vessel that any intrusion below the countertop might interfere with vanity storage. Draw functionality, etc.

One sink versus two? If the bathroom is small, that might lead you to one sink. One person washing at a time, more counter space, etc. Plus, with your budget being tight? One sink will save you.

If you think you'll have trouble renting this place and you think you might need an edge, or if you're renting to a couple/family, then two sinks could give you an edge in a tight rental market. Sometimes it's all about the bling.

Just make sure that before you go the two sink route that the vanity itself can take two undermount sinks, or two drop-in sinks, PLUS the plumbing, without completely losing the functionality of the top bank of drawers in the vanity. If an undermount will cause you to lose storage, then I'd minimize the amount of lost storage by going with one sink.

Ditto the above. If selling, two sinks would be more important, and I would rather have an undermount, but I'd much prefer a sink at a good height than one that is too low. With a solid surface counter, the splash would be less of an issue than with some surfaces -- just make sure the faucet is well chosen for the sink and installed so that it doesn't splash or cause endless splashing when washing your hands, etc.

Exactly what mongo said. It's a rental so go with an affordable option. Check if the vanity can fit an undercount or drop in. Also make certain it's high enough. I've seen some vanities made for vessel sinks that seem lower than normal. I think one sink would be just fine especially in a small bathroom. Though sometimes two sinks might give an edge for a family or multiple roommates. What is the space plumbed for now?