You do not need access to the old email service itself unless Apple is trying to send some kind of verification message to it. We're working with a dearth of information here and I don't want to have to post links to articles covering every single possible scenario until I know what you have tried and what the exact problems are. For example, in your topic line you say you forgot the password but then do not include this in your problem list in your post.
Click on this link to read what to do "If you forgot your Apple ID password" --> https://support.apple.com/HT201487
If you have not used this account in over a year it is possible Apple may have deleted it altogether:
https://www.apple.com/legal/internet-services/icloud/en/terms.html
VII. B " In addition, Apple may terminate your Account upon 30 days’ prior notice via email to the address associated with your Account if (a) your Account has been inactive for one (1) year; or (b) there is a general discontinuance of the Service or any part thereof."
Of course this email would have been sent to your college account and emphasizes the importance of keeping account information up to date.
So, first try to log in using your old information and see what happens.