Since you say you're considering cycling because of the budget stress of the bus pass increase, I'm going to assume that you don't have a lot of money to throw around on expensive clothing and gear. And in my opinion, you likely don't need it anyway: 4km isn't that far and 0Celsius ain't that bad. You should be able to get by without special equipment. Your bike should work just fine and you probably already have clothing that will get you through the winter. Most people already own clothing appropriate to the environment they live in. If you need more, I'd suggest frequenting secondhand stores. You should be able to find the basics there immediately and with patience some real gems might turn up. Today I scored some high end Gore-Tex clothing for insanely cheap.

If you can free up some cash I'd prioritize getting some bike lights (if you don't already have any) and some basic rain gear (if you don't already have it) and some sort of gloves and ear covering hat (if you don't already have them). Dress in layers. After a few commutes you will start to get a feel for how much you should be wearing. Add or subtract layers as necessary. Fenders will make your ride more comfortable if you can afford them. Where are you? Is there a local bike co-op in your area? Those are a real boon for finding cheap bike equipment.

The good equipment costs and does improve the comfort of the commute but it is not necessary. When I started bike commuting I didn't have a lot of cash but I got by just fine. Over the years I have gradually upgraded my equipment and if you stick with it you might do the same. For now, it will mostly be about mental attitude. Give it a try and see what happens.

Best of luck, and keep us updated.