Renter's practical answer, having done exactly this in two flats.
Start with the draught, because it is free to test and it is usually most of what you are feeling. On a cold day run the back of a damp hand slowly around the frame, the sash join and the bottom rail. You will find the gaps immediately, and often the worst one is not the window at all but the gap under the door to the hall.
Self-adhesive foam or brush strip on the sash costs very little, comes off cleanly if you buy the removable kind, and in an old wooden window it can be the biggest single improvement per pound. Do that first and see what is left.
The other cheap thing nobody mentions: if there is a radiator under that window, make sure the curtain does not hang over it. That arrangement sends a good share of the heat straight up the coldest surface in the room and out. Tucking the curtain onto the sill is free and takes five seconds.
And keep every receipt and every original fitting. Reversibility is the actual constraint in a rental — a fix that improves the room and costs you the deposit is not a fix.