I have a staph infection in my finger that won't go away. My eye is bleeding and also out of my bottom. Is that related?

Bleeding. Staph infection in finger is probably separate unless there is some underlying immune deficiency. The blood in eye may be subconjuctival hemorrhage and bottom hemorrhoids. If this persists or recurs obtain lab work at your doctor's office.
Possibly. With your description, you need to go to the er asap for urgent evaluation. You may have a systemic infection with resultant clotting issue causing bleeding. It may also be unrelated and coincidental (an infection with a speeder eye and GI issue) but you should not take that chance.