Can I put my iPhone 4s sim into an iPhone 5s?

Like my contract is nearly finish now for my iPhone 4s but muffin brought me an iPhone 5s free sim. So could I just swap my sim into that an carry on with my contract?

No. Not just a simple swap. The 4s uses microSIM & the 5s uses nanoSIMs. You either would have to cut the SIM to size or get a nanoSIM with your number on it to use.

No, they use different SIM cards. You can get a free replacement from your carrier though.