I have an Apple display (no thunderbolt) and an N64:
I don't have a TV. Is there any way I can hook the Apple monitor up to the N64?
Not easily (or on the cheap). That display has only digital inputs, while the N64 only outputs analog video signals. Therefore you need a converter box.
These come in many different shapes and sizes, generally the more expensive they are, the better the quality. They convert analog inputs (composite, or better S-Video) to HDMI or DVI output.
But wait! It's worse still. The particular display model you've linked has no more HDMI or DVI inputs, only Thunderbolt. Which means you need another converter box to translate the HDMI signal to Thunderbolt format, such as:
Altogether you're probably better off (and spend less money) with getting a TV or monitor with analogue inputs.