Leaving a Part Time Job Before I Start?

Ok so I recently managed to get a part time job at Sainsbury's this is my first ever job. The problem is my contracted hours are 16 hours a week 8 hours each on Saturday and Sunday but I have uni exams starting next week. In my interview I mentioned that I had exams coming up and that I'd be looking to take time off for them and I mentioned this again in my induction and was told that someone would phone me about it. Nobody phoned so I just called them right now and my manager is saying that he has no knowledge of me needing time off. He said he can't give me the next 2 weekends off because it's such short notice and it's busy because obviously it's Christmas. So I asked if I could do reduced hours instead and he said he needs to check and see if people can cover me and that he can't guarantee anything and he'll phone me back later. I really need the time off to revise for these exams I have exams on 5th, 8th and 11th so I don't want to be working all day Saturday and Sunday then taking an exam on the Monday that's just stupid. I clearly mentioned I needed time off for exams in my interview yet they still employed me and now I'm in this mess. If they don't compromise should I just resign? I really do want a job but I have no experience and it took me ages to get this one but I need to study for my exams. I'm worried that if I leave then it will take me another 4 months to get a job again. Any advice would be appreciated.

This does sound like a frustrating situation and there is a lesson to be learnt there which is to get things in writing and talk to your manager about them as early as possible. This happens with bog companies where there are lots of employees and people move around and leave a lot so things don't get recorded or passed on to the relevant people. It is important to revise but as you say you want to job as well. Your studies should come first at this stage unless you plan to have a career in Sainsburys now (you can always come back later). I would try and organise myself to get the revision done around your shifts but if not then resigning may be the only option unfortunately. Make sure you sit your manager down and have this conversation in person so that you don't leave on a bad note.

If I resign will I have to put on my CV that I worked for Sainsbury's even though I never actually worked?

Nope

Also, if I resign will I still have to go in tomorrow and hand in a letter and will I still have to work tomorrow? It says in my contract I or Sainsbury can terminate the contract at any point in first 4 weeks of employment.

You will need to write a letter to resign and it's always good to have these things on record even if you email the actual letter across after scanning it. They may ask you to work tomorrow and Sunday and if you can it is a good idea to do so as it will leave a good impression.

He's just phoned me back and he can't change my shifts. I think I'm going to have to resign. Should I phone him and tell him I'm resigning or should I go about it another way?

Like I said before you need to do this nicely so ring him and talk to him about the situation and then follow it up by email.