The whole process turned out to be incredibly easy. As I indicated in my original post, I had the requisite letter from my hotel, which turned out to be all I needed.
Last Thursday, I printed out the Visa application from the Russian consulate website, picked up a $200 money order from the post office ($200 is the fee for next-day service), and picked up three instant passport photos on Lexington Avenue before heading up to the consulate on 91st Street. When I walked in, there was only one other person in there and no line. The only hitch I experienced was that they did not like the way my application had printed out, so they handed me another one and I had to fill it out again, which took about 10 or 15 minutes. Then I went back to the window, gave the clerk my application, the letter from the hotel, one passport photo (that's all they needed, even though something I read indicated they needed three), and my passport. She walked away and came back in two minutes with a receipt, and sent me over to the cashier to pay the $200. After paying, I brought the receipt (now marked "paid") back to the original window. She gave me a copy and told me to return the next day after 11:00. The whole thing took less than 5 minutes, excluding the time it took to re-complete the application.
The next day, I showed up at around 11:30 and had my passport (and visa) in hand and was out the door in less than 30 seconds.