Arduino project - will this fit together?
From what you've provided everything seems to be compatible.

Concerning the battery, a quick look at the Arduino website shows:


  Input Voltage 3.35 -12 V (3.3V model)


Meaning the regulator will work as long as the battery provides voltage above 3.35 but since your battery's voltage drops as it gets depleted you might not get the battery life you'd expect :/. In other words: it'll work initially  until the voltage drops below 3.35V and then you would have to recharge it.


  Is using Li-Ion in any way different from regular AA


Yes and No.
No : both provide a stable DC source and both can be used in similar applications.

Yes: Li-Ion batteries can be recharged and have a longer life-span, and most importantly, Li-Ion batteries don't have a sharp voltage drop as they get depleted, unlike regular AA batteries. Which is a bonus for this project ( i guess :p )


  I see 3 pins on the back of it


There are only 2 pins that actually provide the power from the battery while the third is usually used to monitor temperature or battery life remaining.So you can use this battery by only connecting the power providing pins (usually indicated by + and - ) and safely ignore the third.

Gd Luck!