First of all, I would check whether the neck is straight.
Place a ruler against the frets (the edge of the ruler against the frets) and if there is any space between any of the frets and the ruler, the neck is not entirely straight. If there is no access to any sort of truss rod adjustment, then you will probably have to put up with the high action. If there is a bow in the neck, the action will appear higher at higher frets. If you lower the saddles in order to fix that high action, you will get fret buzz at the lower frets.
If the frets are straight, try adjusting the action at the bridge. Nearly all basses have a setting somewhere on the saddles that allows height adjustment. Once you have them a comfortable height, or the lowest you can without any buzzing, you will need to check, and possibly, fix the intonation.
Tune the guitar to pitch using a tuner. Play the 12th fret harmonic on the string, and then the 12th fret note. In short, the rule is:
If the fretted note is higher (sharper) than the harmonic, then the length of the string needs to be lengthened. If the harmonic note is sharper, then the length of the string needs to be shortened.
The way you adjust string length depends on the type of saddle, but most cheapo basses usually have a Phillips screw of some sort that you can adjust.
There might be a problem with the nut, but I don't know a lot regarding them.
In this case, I would usually recommend the instrument be taken to a guitar tech. They usually do a fantastic job on fixing the instrument's little niggles straight (and your bowed neck if necessary), you will be able to hear and feel the difference when you get it back. Depending on the service, it may be a tad pricey (£40/$64) but as you say the bass is pretty cheap, so it shouldn't hurt too much anyway. You can always fix further issues yourself, but whenever I happen to acquire an instrument I always like to have it perfect and go from there.
Hope this helps :)