Can I get an STD by sharing a cigarette?

NO. Generally speaking, no. But if the person you share cigarette has cold blisters on his/her lips or anywhere in the mouth, (herpes infection (type1), you can get herpes from the person.
No. Stds are not spread by that small of amount of saliva on an inanimate object like a cigarette. They are spread by infectedbody secretions but saliva has far too small a volume to spread stds.