A fuel pump problem?

1992 Mitsubishi Galant, 5 speed manual - twin cam 2.0 liter, mileage: 83.000 kms. New spark plugs, cables, fuel filter. The problem – It doesn`t start right away in the morning but until trying several times (4-5). Once it starts, the rpm goes up to 2000 and, instead of declining little by little to 900 rpm, it drops suddenly and it reaches down 400 rpm with the tendency of dying. This behavior also occurs during the day when the engine has cooled down. Also, the engine jerks o hesitates in first and second gear when it´s accelerated between 1000 and 2500 rpm. In other gears and at a faster speed, the symptom is not noticeable. The engine looses power when driving on a hill going at a low speed (20-30 kph). Could this be a faulty fuel pump or a combination of two faulty systems?