If I have something lodged in my conjunctiva, how soon should I get it checked/fixed?

ASAP. Dont wait. It can become embedded or infected and therefore harder to treat.
After several days. The conjunctiva and supporting eye structures are terrific at cleaning themselves out. After all they are bombarded with stuff all day whenever you are walking around or when the wind blows your way. If something is casually lodged on the conjunctival surface, it will almost certainly clean itself out. If the object struck the eye at high speed, then get help.