Mine are on my forearm but I can imagine it would hurt a lot under the arm. It hurt a lot on that wrist-bone thing. Anywhere with bone or sensitive skin it will be bad, but like KROC said you have to just suck it up.
Don't forget to take time letting it heal. No direct sunlight, large amounts of water/soaking, and just LEAVE THE TAT ALONE!
No matter how crappy it looks with flakes of colored skin clinging to your arm, just leave them alone. Add more A+D or lotion, never EVER pick them off. I hate the whole healing process but I used Tattoo Goo for my last couple and it worked really well.
You can get some there.
Also even once they heal do keep out of direct sunlight or apply non-fade stuff (you can get that from the site also).
You may know all this anyway, but I like to put it out there just in case. A kid in my class once had just got a tat and he let it "heal" by just drying out and flaking off. It was black and red and in like 2 weeks it was grey and pink.
Tattoos are with you for life. Do take care of them as much as you can for the first couple weeks and even then the next couple months also.