How does anyone cope with it? They just do the best they can. AFAIK it's impossible to make jet lag go away quickly (I've looked online for ways a few times).
As far as I know though, top tournaments are usually months apart. So they can arrive very early if they'd like. I've heard of Anand doing this for example.
Very often top-level grandmasters immediately leave one tournament and travel across at least a few time zones to another tournament. How do they cope with the fatigue?