No, the battery you plan to use will not be able to supply as much power as you want (unless you exceed the manufacturer's recommendations). You want 5W out of the power shield. With a 3.7V nominal battery voltage that means you need at least 1.35A from the battery itself...probably 1.5A after taking the efficiency of the dc/dc converter into consideration. That's a discharge rate of 3C for the battery you have selected, 3 times larger than the manufacturer's recommended current. You should probably look for a battery that can supply 2A continuously. Also, the power shield is right at the edge of it's capabilities if you ask for 1A continuously.
You can't "amplify" current and voltage from a battery. Power is conserved, so if you want 5W of power out you must provide at least 5W of power in. You can increase the voltage with a boost converter but you will get less current. Or, you can lower the voltage with a buck converter and get a larger current. In other words, there is no circuit in the known universe that can take 3.7V and 500mA input power and give you 5V and 1000mA output power.