If it were later in the morning I'd recommend the N or W trains to Astoria Blvd, then look for a taxi (or gypsy cab, which are usually beaten up early-90's Town Cars). You can also catch the M60 bus there (free transfer from Subway with metrocard) when it's running.
But if you really have to be at LGA by 5am, I'd just suck it up and pay for a taxi the whole way. Most buses (not sure about the M60) don't even run that early in the morning, and the subways may only run once every 20-30 minutes. Plus there's unlikely to be any traffic, so in the time you spend waiting for the subway or bus you could probably just about get to LGA. That is well worth the extra $15 or so that it's going to cost you to take a taxi the whole way instead of the subway/taxi combo.
It's also definitely going to be easier to find a taxi in Manhattan at that hour than at the Astoria Blvd stop. The last thing you want is to arrive in Queens at 4:45 and spend 15 minutes looking for a cab to take you the last leg of the journey. Sleep in an extra half hour or so and just take the taxi.