I want to try and go over to Oxford for a couple days in July, but
I'll only be able to go if it's real cheap (college-student budget),
like less than $800 for flight and hotel. Is this even possible?

Using a combination of Mongolian and BA, you get a return from Ulaanbaatar
for around 1200 GBP. Please don't keep us in the dark about where you are
departing from??