Yes it's really easy. New chain will not necesarily be the same length but should have the SAME number of links as the old one which you should count to make sure. Thus it will be shorter as it will not have stretched so much. You will need a pair of chain link pliers, Park (MLP-1) are the best and worth every penny. It's one of those tools you can't do without but saves you no end of cursing if you don't have a pair. Also NITRILE gloves as chains are oily and messy. I use SCAN deluxe oil and grease resistant. You will also need a chain splitter which could well already be on any multi-tool you might have which you will need to shorten the replacement chain as they are invariably longer at 114 or 116 links. You need to get the correct number of links in the new chain matching the old one. And to monitor wear in future it might be worth getting a chain checker. Again the Park tool CC-3.2 for £8.99 is all you need. I would have thought the cassette was still fine with many thousands of miles of riding left in it. If you ride with a chain that is knackered or devoid of lube then you will cause premature wear of the cassette and chain rings.
So in summary you will need,
chain splitter
chain link pliers (Park MLP-1 for removing a quick link which you have fitted to your replacement chain)
chain quick link either SRAM or KMC (get the right quick link for the chain eg 8,9,10 or 11 spd chain).
Nitrile gloves
Chain checker (Park CC-3.2) to monitor wear of replacement and future chains.
Quality chain lube. Do a search as there has been gallons written on this. With new chain lube it before you put it on.
CRC have all the above.
It is also worth thoroughly cleaning the cassette and front chain rings when fitting a new chain.
If doing this inside or even outside put a lot of paper or old sheet on the floor/ground to prevent oil stains getting you into trouble with OH.
HTH.