Devonai makes a good point. How accurate would an arrow be in a vacuum?
More accurate than in air. True, no fin stabilization, but also no aerodynamic negative effects, so you'd be trading imperfectly corrected aerodynamic effects for no effects at all. Winner: space bow
As for the OP question, without magazine fed, repeating firearms, the Mongols win.
Let's look at the stats.
First, can we stipulate that the Mongols begin 400yds away, and also stipulate that if they close this distance before losing more than 15,000 of their number, (2:1 ratio cavalry vs. Foot soldiers at lover range? Cavalry wins).
Okay, the math. A horse at full gallop travels at about 30+ mph (not a racehorse remember), or about 44fps. To cover 1200ft, it takes about 27 seconds. The 2400 mexicans therefore need to kill 15,000 Mongols in 27 seconds, or roughly one Mongol per Mexican every 4.5 seconds. Given a typical single-shot hit percentage on a 30mph moving target of 20% increasing to 80% at near zero range (giving the defenders the benefit of the doubt), or an average of 50%, the defenders would need to fire an average of 12 shots, accurately aimed, at a rate of one shot every 2.25 seconds or faster.
Given the above, I would conclude that without WWI level weaponry at least, the Mongols win, in fact, I think even a WWII infantry force would be close to losing
My logic:
WWII US weapons: M1, M2, browning 30, and the BAR
M1: 10 shot stripper clip...assuming 4-5 seconds to reload and re-aim, 12 aimed shots takes too long
M2: infantry emplaced tripod had limited arc of fire, can't traverse 180 degrees in less than 27sec and then fire the 30-50 rounds needed (3 man crew, thus 18 kills, lower hit percentage means 72+ rounds needed, so the traverse and reset has to take less than 20sec)
Browning30: same issue
BAR...now it's possible, 20rnd magazine, but low issue rate means minimal overall impact, and "oh crap" realization will likely reduce effectiveness of aimed fire, and again, reloads take too long.
Basically, without modern repeating, large capacity firearms, AND vehicle mounted rapid traverse crew served weapons, it's the Mongols...or it's damn close (and I'm really giving the defenders the benefit of the doubt in hit percentages on a horse mounted rider...cut the hits down to 20%, and even with modern weapons, it would be the Mongols (let's see someone hit 6 of 30 against a jinking mounted rider with an M4/16).
What stopped the use of cavalry and human powered weapons wasn't the individual firearm (at least against a threat who didn't care how many died, e.g. The Mongols in this case), it was artillery with grape or exploding shot and/or large capacity crew served weapons.