It's no news in Nigeria that working in a bank is very stressful and comes with little or no satisfaction. For a graduate of banking and finance, where else (besides banks) can He work with no experience (i.e entry-level)? Excuse my ignorance as well, but what other career options are there for a graduate of banking and finance besides being a banker?

Also, is it advisable for students who intend working in the financial sector and are yet to gain admission to the university to apply for banking and finance? Are other courses such as accounting or economics better? Where can entry-level graduates of these courses work as well?

Once again, pardon my ignorance.

There's nothing wrong with Banking and Finance as a course. You can work in almost every industry as a Finance professional. Whether its Banking and Finance, Accounting or Economics, work experience and professional certifications will define your career path. Infact, your first job is a massive decider to their future career path.
The relevant professional certifications are CIBN - for Banking, ICAN/ACCA - Accounting, CIS/CFA - Stocking Broking and Investment Banking.