If you’re going to be here for a year it’s probably not worth it to get an apartment because it means having to patchke a bit more, which might detract from your vacation experience.
I’d recommend a bus tour of Chevron, at the very least. You may want to do others especially if you don’t have a car. Public buses are a pain. It makes it much easier to get around if you have an organized group.
Do you want the big sites (especially if you haven’t been here in 20 years)? Or are you looking for just small cool things to do?
A cool thing to do in general- Temple Mount Sifting Project. A pain in the neck to get to by bus but pretty ok by car, it’s in a gorgeous park next to Har Hatzofim. 20 shekel for two ridiculously cool hours of sifting through dirt from Har Habayis for an archaeological dig. Until you’ve found a Bayis Sheni lamp in a pile of pebbles you can’t fully understand how cool that is.
Highly recommended.