Hello. While I have happily used the Express and Connect trains, in this case, I would choose the Tube, Piccadilly Line. A flight that seems to match your times and cities looks to arrive at Heathrow, terminal 3. Google it. In any Heathrow terminal, there will be information kiosks with people to help you. They will help you get the necessary ticket for the Tube. ATMs will be in the terminal, as well.
Get yourself a small foldable map with some sights on it - or a small handy guidebook. Leave Tube at Green Park or Piccadilly Circus (I would choose Green Park, myself), plot your way (walking) to Westminster area.
Eating: Google (maps): Pret a Manger, Green Park, London. This will show some locations of the eatery and sights you may be interested in. Print out what looks interesting. You can even plot your walk to Westminster Abbey, or other location you may be interested in. Think about necessities like toilets. For example, Fortnum and Mason is in the area, and, in addition to being an interesting place to visit, you will be able to locate toilets there as in any department store.
I think Westminster Abbey is a good location because a) perhaps you can get to tour the abbey; b) there is a nice cafe there (and toilets) - even if you do not tour; c) at 2:00 in the afternoon, you could hook up with the London Walks Westminster tour. Walks take 2 hours and are well worth the minimal cost. Their website has directions and maps of where to meet. At the end of the tour, repeat your steps and ride to Heathrow.