Check your housing contract to see if firearms are allowed on the landlord's property. If not, then file a complaint with your landlord and they will take care of it. If he refuses, he is in violation of the housing contract and may be evicted. Also may not be a bad idea to have a talk with local LE about the situation if you think this guy is serious. If your concerned about your computer, look into some basic encryption software for any sensitive info you have on there or encrypt your entire drive.
There's also key logging software you can install on your own computer that will monitor the directories and such so you can get an idea what he's trying to do