Can my apartment's electricity harm my computer? Is there an easy way to test the electricity?
Yikes, it sure sounds like you have several issues with your electical wiring there. I won't speak too much to your power supply issue, but really think you should invest in a UPS to mitigate any power related damages to your equipment.

Now lets correct a few mistakes in the previous answer. 

The 120VAC legs of your electrical service are supplied by two separate hot feeds from the power company. The neutral is not there to split the 220VAC, but rather to supply a return path to the service panel and ultimately earth ground. 

The neutral should be at the same potential as ground. An open neutral is checked by measuring neutral to ground, if line voltage appears, then in fact the neutral path to ground is open. Unless you have a situation where both 120VAC legs are in contact with one hot connection, you cannot produce more than 120VAC on any open neutral. If you were to see 140VAC on any hot to ground or neutral,  or neutral to ground, then you have some serious problems and time to call the landlord before you fry all your lights, appliances, and electronics. This situation is very rare, and I don't think it is your problem.  Your problem sounds more like a bad connection, either hot or neutral in some device that feeds your downstairs outlets. 

This can be a frustrating and time consuming job to track down this type of problem. It may involve pulling several devices out of the boxes and testing voltages with the power on, and also testing conductivity with the POWER OFF. Unless you have the experience and you feel very comfortable working on hot wiring, I'd suggest you have the landlord call in an electrician ASAP.