I've got one of these:
Would I just upload the Adaboot onto the Arduino using the ICSP header on the Arduino?
Is there a downside to using the Adaboot?
I have not read about any downsides. I've not used her bootloader so, once again, I cannot give you a straight answer.
Like would I only want to do this to the Arduino I plan on embedding, or is it an all around better bootloader?
It's my understanding that there are few or no issues with it so I suggest considering it a better choice. I also suggest modifying only a single Arduino until you've had time to decide for yourself if you prefer her bootloader. If you modify all the ones you have and run into problems, it may get a little frustrating having to put back the original bootloader!
If you're considering using a "better bootloader", you may want to have a look at Optiboot...
(Why isn't it standard if so)
I believe the Arduino folks are moving towards "Optiboot". I believe it incorporates Lady Ada's modifications.