Cville has felt safer for me than many places I have been, although I'm white so I probably can feel safer because of that. I've also been making due without a car for anyone's who's interested. Public transport is pretty bad and the hills can be intense, but between walking and biking I can do what I need to do. No weekend hiking trips but who's got time for that.
In terms of neighborhoods, the ones closest to grounds (Along Jefferson Park Avenue, or JPA) and just north of grounds (South of Grady) tend to be full of undergrads. Grady in particular is frat row. Fifeville is full of a lot of grads/med students/young professionals, and it's not a horrible walk (~30 mins depending) to grounds. 10th & Paige (South of Preston, North of Main, East of Grady near Venable) has fewer undergrads and is a shorter walk (~20 mins) while also being close to downtown. I also know people who commute from "downtown", which is more of a 45 minute walk but you could take the Trolley, a free bus route that is fairly reliable.
It might not be impossible to find a studio or 1bd for 1k, but it will be harder. 1.2 k is more likely.