As the other 2 posters have said, Ann Arbor is a liberal city. I completed my undergrad at U of M and lived there for 4 years. I spent 2 years in the dorms and 2 years in off-campus housing. You didn't specify, which graduate housing is he in, Northwoods or Munger? I'm assuming Northwoods since you mentioned lack of A/C.
It's a weird situation regarding firearms on campus. Michigan has preemption for firearms laws meaning that local municipalities cannot enact stricter firearms laws than state laws. Yet the big 3 state-funded universities (U of M, MSU, and Wayne State) all prohibit firearms on campus and campus owned facilities. This was challenged in court, but has been upheld so far. At this point, he cannot possess a firearm in U of M-owned graduate housing, nor can he have one on campus. This will also apply to you visiting and moving him in.
Other than U of M, MI will recognize CCW permits from other states as long as you are a resident in the state which issued your permit.
The only local gun shop I know up there is Ann Arbor Arms.
There are plenty of great restaurants in downtown Ann Arbor, but not so much on North Campus. I enjoy Blue Tractor, Jolly Pumpkin, Zingerman's, Blimpy Burger, Tios Mexican, and many more. The best area for restaurants will be along E Liberty St from State St west to Main St and along Main St from E William St to E Huron St.
There are Home Depot and Lowes. There's a hardware store up near Plymouth Rd and Huron Pkwy that should have window units. Otherwise, there's a Home Depot a ways south at Carpenter Rd and Packard Rd.
As for hotels, if he's in the Northwoods apartments, you'll want to stay up at the Hampton Inn, Holiday Inn, or Residence Inn at Plymouth Rd and US-23. If he's down in Munger housing, you'll want to stay at the hotels near Briarwood Mall.