According to
EC-131 (Fail-safe Chart) and
EC-325 it doesn't seem that a P0335 code will cause your engine to fail to start. But you can try following part of the diagnosis process - at least verify connections at the sensor to try to solve that.
You should measure your fuel rail pressure with a gauge or at least feel the fuel lines to see if you are getting enough fuel pressure. You might be able to check at the fuel filter's rubber hose for pressure, I don't know where the fuel lines are in the engine bay either and not sure where to look in the FSM to find them. See
FE-6 for instructions on replacing the fuel pump if you decide to replace it.