The MRK1000 says it is 3.3V and would need bidirectional communication to do 5V.
No you do not need bidirectional communication, you only need a single direction level shifter. The link above is not designed for neopixels but for I2C communication, which is much slower than the speed the Neopixels communicate at. Anecdotal evidence suggests that this sort of board is marginal at best with people reporting it stops working when fingers are placed on the circuit board.
Is it possible to use this arduino for this type of project?
Yes.
I use a two gates from a single 74HTC14 chip to boost the logic signal to 5V from a 3V3 system. The circuit is so trivial that I don't think anyone sells them on a ready built board.
Put them in your parts box for other projects. These indeed won't work for the ~900 kHz those neopixels (which are really just WS2812B LEDs) communicate at, they are great for I2C level shifting.
No experience with this one. It's bidirectional, which you don't need here. The 74HCT14 is probably the best solution here, and those chips come at mere pennies a piece.
When I search for that number in Google the whole of the first page is filled with links to various data sheets and manufacturers of the hex Schmitt trigger inverter, which is what you want.
By the way, what is so grand about a MRK1000 WiFi? Very expensive for the functionality.
If you want WiFi, a WeMOS D1 Mini is a good "go to", very powerful and dirt cheap (which means, you do not just buy one, or two)! If you do not want WiFi, a Nano will drive NeoPixels directly.