One of the first things to think about deals with issues every house or building has: plumbing problems. While many can be simple fixes, sometimes they need a little more attention than your tenant- or you- can give it. While this is possible, try different routes first. Check that your tenant has a plunger in case the toilet or sink becomes a little backed up. If they don't, it's always a good idea to keep a cheap spare in a storage closet. Another good thing to keep on hand is a disposable, single use snake. Although they are not long, they can definitely be useful for small clogs that are shallow in the pipes. Also, you should keep a tool box with these things which are used only for the apartment or building just in case you forget something there. This way you won't lose all your tools and hardware for good. If a problem proves to be beyond your abilities to fix, make sure to supply your tenant with the number of a good plumber that you trust just in case you can't get back in time to make the call or if it's an emergency.
Another major issue buildings often have deal with electrical faults. The electricity in a building can be finicky, especially if the wiring is older. While every building has its own quirks that residents need to learn, even the most up to date homes can have issues which are beyond human control. One option that you can take is the purchase of a generator to keep some power running to the building in the case of an outage. Even if it only provides emergency power to a couple of outlets, this is always a good idea, especially if you live or rent in an area that is prone to severe weather.
While no one expects you to be an all-around handy man, there are some things you can do to help ease both your and your tenants stay in the building. These may turn expensive, but they will not go unappreciated and will certainly pay for themselves eventually in the long run.