I arrive at terminal 3 Heathrow at 11:00 PM. Do I have enough time to clear customs and catch the last airport express train (11:38 PM) to terminal 4. Can I walk it? Other options?
For complete Heathrow connection information, use this page on the airport's website: http://www.heathrowairport.com/heathrow-airport-guide/flight-connections/connection-guides. Put in all your information, and you'll see the step-by-step directions and estimated times for any connection.
I would take a taxi. It won't be real cheap, but worth it.
No.
Are you just going from T3 to T4? Are you staying at one of the hotels there? If you are staying at the Hilton, you might contact them for options. There is a slim chance you could make it (but very unlikely). On my last flight to LHR, my flight was wheels down at 9:43pm, at the gate at 9:49pm and I was at the tube station at 10:23pm. That is quite quick, but I've yet to encounter a long immigration queue on my late night arrivals at LHR. If your flight lands early, you might have a chance. Late night arrivals are tricky, and there is no walkway between T3 and T4.