Hi, I'm really familiar with this problem. In the LA area, most radio stations are owned by one of nine companies and most are clones of each other, so we have one good FM station and one good AM station. I think that you'll find most any radio built in the last 30 years will have lousy AM band support. Good AM support can be found on most any component stereo from the early '70s and '60s. These can be had cheap from second hand stores.

As for the antenna, the AM band wavelength is something like 200 meters, so the smallest resonant antenna would be 50 meters long. Obviously that ain't gonna happen in an apartment, so just run a small wire as long as you can get it. What I did at my apartment is throw a tiny wire from my balcony to a near by tree, and then threw it from tree to tree and let it end about 100' away. In the end it was really hard to see if you were looking for it. This worked really well with an early '70s vintage Pioneer stereo.

Good luck!