Media remains 'local' to you computer - unless you are on a network or server. It's only licensing, apps and sign-in that has moved to the cloud.
You install applications from Adobe by downloading the Creative Cloud app and then choosing what software to download. Which is still run locally on your computer.
Your 2013 laptop may well not be up to the task of running Premiere Pro. Adobe have a list of specifications required. Check them out.
However, you could also run a 'trial' version of Premiere Pro at no cost and see how you go?
Updating your machine and using FCP-X could be a great option ... but if you did splash out on a new Mac you may 'still' decide to go with Premiere Pro. I think FCP-X is great but it's a different beast and not like the FCP-7 you may have used in the past. Premiere Pro is has much more of the 'traditional' non-linear editor workflow and may be easier for you to get back into.
So I suggest get the trial, play with it on your old laptop (even if it is not going to run perfectly) and use the trial to decide which way to go with new hardware.