If nothing else works, power conditioners exist. That wouldn't address your lights, but it would keep appliances plugged into it happier. Some types of UPS also have this buffering behavior.
Some types of LED bulbs may be less sensitive to brownout than incandescents are.
But the place to start for a real fix is your landlord, and the landlord's place to start if they're too cheap to call an electrician immediately is the power company.