Completed application form
Two passport photos
Travel and hotel bookings
Copy of my passport
I had to pay USD 58.00 in cash and it took only two days. The whole process was very easy.
Note Google maps locates the consulate slightly off if you type in the English "Mongolia consulate", but locates it correctly in the Russian version (Posol'stvo Mongolii V Almaty), very close to the Hungarian consulate as Caravanistan says. Yandex seems to have it way off so be careful if you're using that taxi app.
Ring the bell at the gate and someone will let you in.
