The size of shot glasses varies greatly but the average shot is 1.5 ounces; and there are 8 ounces in a cup. Thus, you'll need about 5 shot glasses to fill a cup.
You can also buy "short shot" glasses or "pony shots" which are 1 ounce. Pony shots are commonly used with martinis, Manhattans, and Rob Roys.
It's worth giving some thought to your own pattern of drinking and how you can manage it so you enjoy alcohol safely. Check out some tips for managing your drinking, shared by Know Your Limits.