Does the Army Reserve try and bump you up a rank when switching from active duty?

It depends on needs of the unit. But I would lean towards no. Because why should they promote you just for coming to their unit and not promote someone that has been there for 6 months more than you.

It is possible but never guaranteed. They have the ability to slot prior service personnel into a position one skill level above their current level (a 30 level service member can fill a 40 level slot on the books). That being said, you still have to perform well enough to warrant the promotion. A lot of Reserve units are hurting for prior service personnel so it is a definite possibility. The worst they can do is say no.

Never heard of a reserve unit bumping anyone up a rank just because they came from active duty, now if have all of your courses etc and can be promoted and they have a slot, I could see them putting someone in that slot and then promoting them.