Nothing wrong with party cups, most of us use them. The light you ordered will probably work for seedlings, but not really best thing you could’ve purchased.
If you had something like a multi way bulb splitter and daylight or cold white (usually around 5000k-6500k) cfl’s will do it. I would look specifically for the 23 watt variety or larger. 3-5 of those should keep plants alive and well for a few weeks. You could do same with led bulbs and diffusers removed.
If you’re into the DIY scene, you could also source higher end led modules from electronics distributors.