It is probably cheaper or the same to just use a pay as you go oyster card than to buy any travel pass on the oyster card if you only have 4 days.
There is heathrow express but that is more expensive
https://www.heathrowexpress.com/
you can also check out heathrow connect that takes 10 minutes more but works up significantly cheaper but frequency of trains less often
https://www.heathrowconnect.com/
check the websites and see if there are advanced tickets which can run out q fast, the other problem with advance tickets is you have to decide and fixed time and the customs lines at heathrow for international arrivals can vary q abit. If you end up late, you cant use your ticket anymore. if you reach early and wait around you might as well just take the tube which takes 45mins-1 hr to reach central depending on yr destinations. I used to live there for 6 years and made frequent trips back every year so am quite experience with the customs and wont advise for you to prebook your heathrow-london tickets. It will probably be safe to book your london-heathrow leg though since for that you can at least control your time
if there are no cheap advance tickets/you cant decide on the time, can just buy on the spot at the airport. there are many machines where you can buy the ticket from
But depending on where you are going to be based in london,and contrary to popular belief, the heathrow express/connect may not be much cheaper nor faster as well. This is also bearing in mind that you will have to change from the trains to the underground tube if you are going anywhere else that is not paddington is not exactly the most central place in zone 1. Say if you were going to waterloo or the east or south bit of zone 1, you may end up taking similar times +/- 10 minutes and have to factor in the extra time to change to connect from overhead trains to underground trains, and may also require to change the underground lines a few times depending on which underground line your destination is at. perhaps you can advise us on yr destination and we can advise you better.
If you are taking the flight that arrives at 5/6am, usually no issues with the tube then as there is not much traffic at that time in the morning. I always travel on the tube with like 2 luggages at least by myself and have coped with it (just bear in mind there are stairs on many tube stations/overhead stations)