On the process of heating a gas at constant pressure
OK, let's go one question at a time.


  does having constant pressure automatically mean there will be a temperature change and if so, why?


Not quite.  Adding energy means there will likely be a temperature change. What
constant pressure means is that it will be easy to calculate this change, because you can use the value of the constant-pressure molar heat capacity, $C_p$, which is the same for all diatomic ideal gases.  It is also independent of temperature, volume and pressure.  You will find it in your book.

The molar heat capacity tells you how much energy you need to transfer to one mole of the gas to raise the temperature by one degree.  You now have everything you need to know by how much the temperature changes, and therefore the final temperature. 


  What about volume? Why do we not need to know the volume for this equation? 


Actually you do know the volume, or can easily figure it out.  And though the volume is not important for finding the temperature change, for the reason explained above, it's helpful in finding the change in internal energy. Why?
The change in internal energy is the energy you put in (500 J) minus whatever energy the gas spends expanding.  Since the pressure is constant the energy spent expanding is simply the product of the pressure by the increase in volume.  You can calculate the increase in volume because you know the pressure, number of moles and initial/final temperature. 

Now a bit of meta advice.  


  I am trying to understand how to approach this question without simply plugging the numbers into an equation. Can someone please explain their thought process while solving this?


You don't make clear what, if anything, you know (either conceptually or equation-wise) and what, if anything, you've tried. 
You'll get help faster if you show you've done your part.


  Basically, I want to understand what's going on conceptually rather than memorize a few rules.


Understanding what's going on conceptually goes hand-in-hand with learning the rules.  You can't really do one without the other.  (Well, you can memorize the rules without understanding the concepts, but you won't know which ones to apply.)