If you use an I/O pin you don't need a pullup. You write the reset line low at initialization and then you set it high. If you have a spare I/O pin that's the easiest way to do it.
I measured the current on my display at 9.7mA (Vcc=5V) with all the pixels lit. I was driving it with an I/O pin.
On another project I used an Adafruit OLED. For power savings I turned off Vcc, shutting down the OLED completely. But with this Alice OLED I found it doesn't work. When Vcc=0V the display remains on, drawing power from the other pins. The only way I could find to shut it off was to drive all of the pins low, including SCK and MOSI. So I can't use it in the same way as the Adafruit version.