Does this buffer's current rating apply to each individual output, or all of them?
It's for each pin. Note that a datasheet may give maximum values for supply current (Icc) and ground current as well, usually mentioned under Absolute Maximum Ratings (AMR). The 74HC595 for instance gives an AMR of 35 mA per pin, but on top of that a supply current and ground current of both 70 mA AMR. 

The 74ABT244A doesn't mention these limitations, so should be OK with a ground current of 8 \$\times\$ 64 mA = 512 mA, which is a lot! Since this is a higher current device it will use thicker bonding wires than usual in TTL, at least for the Vcc and GND pins.