Assuming the Raspberry Pi is just a consumer of data its GPIO would be configured as an input, and whether or not you enable a pull-up resistor would depend on if that's built into the controller or not. I can't think of a reason why the Raspberry Pi would ever actively drive the data line. I would guess you need to enable the pull-up on the Raspberry Pi, because otherwise what's the point of having an open collector interface on the controller. You might need to use an external pull-up resistor if the internal pull-up is too weak; the tutorial I reference below suggests 2.2k Ohms, whereas the internal pull-ups on the Pi are likely > 30k.
I wonder whether you're actually going to be able to get sufficiently frequent / deterministic sampling of the signal on the Raspberry Pi by polling the GPIO though, in order to decode the (presumably) serial data coming from the controller. I'm not familiar with the N64 controller protocol, so it's hard to say.
I found this reference on the internet, which might be helpful to you:
To understand how to interface with an N64 controller, one must first
understand the protocol that a genuine N64 uses to interface with the
controller. All data is transmitted on a single, half-duplex wire (the
signal wire plugged into pin 25 above). When this wire is idle, it is
high (hence the pull-up resistor). If a falling edge is detected, it
means that bits are being transmitted. Bits are transmitted in 4μs
intervals. For a 0, the wire is low for 3μs and high for 1μs. For a 1,
the wire is low for 1μs and high for 3μs. In order to read a bit, you
must simply wait for a falling edge, and then read the wire 2μs after
the falling edge. If the wire is high, the value is a 1, if it is low,
the value is a 0. All transmissions end with a 1 bit that is not
followed by a falling edge, called the signal bit.
Looks like a serial Pulse Width Modulation encoding. I'm skeptical that one can decode this with a Raspberry Pi easily. It would, on the other hand, be trivial to decode with a microcontroller, e.g. using pin change interrupts and/or hardware timers / input capture interrupts.
Seeing as it is a bi-directional bus, you should indeed never set the GPIO to an output-HIGH. When you want the output to be HIGH, you just set it as an input with pull-up enabled (or no pull-up maybe). When you want to output LOW, set the GPIO to an output low. Again though, I doubt you can get the timing right with a Pi given the O/S in the way.