Does this water heater pipe need repaired ASAP?
Yes, I would replace that right away.  I wouldn't touch it without the proper replacement parts on hand, since you don't know the depth of the corrosion.

Agreeing with other answers, you need a dielectric between the tank and the copper.  Although they claim to be dielectric nipples (T-575 is apparently a dielectric nipple, and I think that's what the label says) I don't see how that is possible as it's directly connecting the copper to the steel.

The corrosion on the steel may also be indicative of the condition of the inside of the tank.