If you're using a smartphone, you can sync your contacts to say, a gmail account and keep it there as a backup. Or borrow a smartphone to get this done. As an added precaution, you can even make a backup of the contacts in the gmail account to a file that you can save to your PC.

Then when you get a new SIM card, just copy the contacts back.

Actually, it's good to sync your contacts to a gmail account in the first place, as it'll be easier whenever you need to change phones, whether an Android or iPhone, the same account can be used for the sync.

It might become more of a hassle not being able to do things like changing a SIM card or phone, so it's good to have a backup somewhere for "just-in-case" moments.

Take your time to study how it should be done, then be careful and don't rush through it.

Good luck!