How to get my humanities paper published in the short timeframe before admissions deadline?
Journal review times in the humanities tend to be longish and I doubt that you'd have a paper accepted in a journal of sufficient quality to impress anybody. (I just got back yesterday a round of comments from a paper I submitted in September.) If I were you, I'd maybe try to submit a paper to a graduate conference instead. Look around for Ph.D. granting programs in your field in the area and send them a paper/abstract. It's much less work for you to write up an abstract, and a much higher likelihood of actually getting something accepted in a reasonable timeframe. An undergrad getting something into a grad conference is probably sufficient to make the committees think you've got potential as a researcher. Maybe it isn't quite as impressive to have a presentation as a publication, but I doubt that'd be a big factor here. To get accepted to a grad program, you just have to get past a threshold of potential, and doing something that puts you way, way above that threshold (publication in a reputable professional journal) just isn't really any more helpful to getting accepted than doing something that just barely puts you above that threshold (a grad conference presentation). 

One exception here: if you have a paper or something that one of your profs was just raving about and told you that you just have to publish it in Such And Such Journal, then yeah, maybe shoot it off, but don't expect it to get picked up in time to make a difference in your apps.