Sorry, I somehow missed that you already got an Arduino Uno at home.
In that case just give it a try, just remember that you have to set the baudrate to 115200 which will limit the number of LEDs and the update rate - about 100 LEDs at 25Hz update rate should work though (haven't tested that myself though).
The ATmega 32u4 based Arduinos don't have that limitation, this is why I generally recommend using them (esp when buying new it doesn't make much sense to spend money on the older Arduinos, like the uno).
Concerning the level shifter: without knowing specific details about the used level shifter it's hard to tell if it'll work or not. Most of the level shifters in the form-factor are passive ones with pull-up resistors and while they may work they are neither recommended nor guaranteed to work for this specific application (unidirectional up-shifting of >= 1MHz signals from 3.3V to 5V).
As the issues can be very subtle and hard to track down ("mostly" or "sometimes" working etc) I generally advise against using such improper approaches.
so long,
Hias