How long will it take for my eoe (inflammation) to go away so i can get off the oral steroid, budesodine?

Your question in not clear. It would be better to check with the doctor who prescribed the medication as s/he knows more about your health.

You should have symptomatic relief within a week but the induction therapy will be 12 weeks. After that, you will need maintenance therapy with topical glucocorticoids and/or dietary restriction. You also need another EGD after the induction therapy. Good luck and take care yourself.