The Devil
A 13 year old battery will be pretty worn out, and so will a Hyundai starter. Well, the whole Hyundai is a junkyard refugee after 13 years. Get an OBD2 code check to see what specific issues the car is having. There is no such thing as "pump starting the ignition", so you'll have to say what that really means. Think about all the maintenance that hasn't been don't on this car over the years. All that stuff needs to be done now, plus a dozen more expensive repairs.
American Patriot
The fuel line can be bleeding down when it sits overnight. Before spending money on chasing a "repair", try this.
When you first get in the car, turn the key to "on" (NOT start and listen for the quiet hum of the fuel pump. The dash lights will be on and the bells will make noise. This, literally, takes 3 or 4 seconds. When the first set of beeps stop, turn the key to START. This delay of just a few seconds, lets the fuel line fill and gives the computer sensors a chance to set. This is ALWAYS a good idea, Too many people jump in a car and turn the key to start the car without giving the car a chance for the computer to set the sensors... and to fill the fuel lines. Pumping the gas pedal in a fuel injector car doesn't pump fuel since it's not a mechanical fuel pump.
Anonymous
Do you mean pump the gas on the first start of the day or pump the gas on every start? If on the first start and then it starts fine for the rest of the day, its most likely your fuel injectors. Any mechanic will tell you the fix cost and also look up the injector problem history for free. I;ve had the same problem on old cars. They do start. Then they run fine. Why blow the money on an expensive fix? BTW On the start, hold the gas flat to the floor and turn ignition for 5-6 seconds. Now let off and start as you usually do.