99 Benz C230 4cyl supercharged

Starts up firsttime just fine and after about 30 min engine just shuts off
try to restart but acts as if out of gas and NO it's not, if the car sit with hood up for about 30-45 min car starts back up untill it heat's up again and problem recurs Thinking ethier it is fuel pump or coil packs on plugs i take temp on coil packs and closeset to firewall is about 15-20 degrees hotter than front coil pack , One other question Is the fuel pump outside of tank what is the long cylindrical thing next to fuel filter under plastic shroud