Translation:I'm bad at singing.
I can hear what you mean, but I don't think it's wrong: the 'a' is pronounced with the tongue unusually far back in the mouth, much like the 'o' in American English 'lot'. However, I think this is probably still an OK Japanese 'a', and it definitely sounds very different from an average Japanese 'o', for which the mouth is not as open.
Just add に onto the end of な adjectives (じょうず、へた、すき、きれい、etc.）.
Is there any sort of rule to use が for skilled/good at and unskilled/bad at?
Not really a rule specifically as you are asking as much as the general sentence pattern of は-が-です, or "speaking of X, Y is Z." You can't have a verb before the が unless it is turned into a noun by adding こと or の to the end of it. The sentence here is a pretty good example of using a nominalized verb with が. That's about the only major rule in play here with が.