Late bachelors graduating overlaps with possible future masters degree start
There are two problems to be solved, and each problem needs to be addressed in the appropriate institution.  Hopefully all involved will be reasonable and a happy solution may be found.  Here are instructions for your friend:


Ask the master's institution for permission to begin the first semester before the Bachelor's is granted.  Explain the request.

If you might miss a couple of days of classes, correspond with the individual instructors.  Explain the situation, and see if you can get the syllabus, text book and first assignment ahead of time, so you can work ahead.
Ask the undergrad institution to allow you to take an early final exam or omit the final exam in each class.  I would start with the dean of undergraduate studies or department administrator or department chair, and then I would be prepared to follow whatever instructions s/he may provide.

However, if there is an uncooperative person at the top, take a somewhat different approach -- start by talking to the individual professors, after you are far enough into the final semester, so they can see that you are a proven quantity.  Explain to each one: "I'm just asking if you you would be amenable to an alternate grading scheme, or an early final exam, in case the department provides approval."  Once you have some instructors on board, then it will be easier to get department approval.

If none of this works, do not hesitate to go higher up in the hierarchy.  If you can, find out something about the administrator your propose to meet with, before you go.  This last piece of advice is not necessary in all countries and in all universities -- but it can't hurt.  If the person seems to be authoritative and negative, maybe skip that rung in the hierarchy and go straight to someone else.


Your general approach should be to be humble and ask for an exception to be made; but if you need to, you can still gently point out that it is an honor to be invited to join the master's program at your-grad-university, and you want to do your alma mater proud.