Licensing for Xbox 360 is complicated, and Microsoft changes it at their whim. Thus, before you follow any advice, I'd suggest reading the current Microsoft support document. You might have to click around to get to the one for your country/region, as the rules change per region. I'd also suggest testing any method with some cheap or free DLC before you commit large sums of money.
With that disclaimer out of the way, here's the solution.
You've got two Xbox 360s in this scenario.
- Xbox 360 #1 is the one with the good internet connection
- Xbox 360 #2 is the one with the bad internet connection
Your friend visits your house where Xbox #1 is. He downloads his Xbox Live profile on the box, which adds his account to your Xbox. He plugs a USB key into the Xbox. He then buys and downloads his DLC onto the USB key using his account on your box.
Then he goes home to his box, #2. Here he can plug the USB key in, and log into his account online. The Xbox will check his account to make sure he has permission to play the game (which doesn't require as much bandwidth as downloading the game). Then he should be able to play his game on his Xbox.