Remember, that Omega doesn't support APs that are n-only, nor does it work at 5GHz. So you need to either slow down all of your other wireless devices by enabling b/g support, or set up a separate AP for the Omega (this is what I've chosen to do). Also, if your WPA uses TKIP, you can't use wifisetup, and need to edit /etc/conf/wireless manually.

Basically, wi-fi - like many other things - is a huge mess on the Omega.
I'm personally regretting buying one more every day, I've already scrapped the project I originally bought the Omega for, and instead used a Raspberry Pi Zero, a Hat and a bunch of other components instead. While the form-factor is bigger, and there are more separate pieces to manage, I was up and running within minutes, whereas with the Omega it took days to just learn why my Omega didn't respond to anything (through support I learned my Omega was apparently delivered with a faulty/missing firmware), and that was only the beginning of my problems, as I moved from re-flashing the board to actually connecting it to something.

Good luck with getting your Omega connected!

As @ccs-hello mentioned, not trivially. But Arduino sketches are just C/C++ programs with some helper functions and syntactic sugar sprinkled on top, so it wouldn't be an impossible undertaking porting a sketch and it's required libraries over to use Omega's GPIO.

Obviously things like power management and timings would be different it would never be a simple drop-in change, but if your code doesn't make use of things like that, it's definitely doable.

All that being said, you would most likely have an easier time of it just re-implementing the sketch from ground up on the Omega, or if you're really married to the Arduino IDE, getting a different, Arduino compatible, board like one of the ESP32-powered ones.

@Bas-Rijniersce Onion Omega2 Documentation - Known Firmware Issues:

WiFi Issues
No automatic support for TKIP-encrypted WiFi networks
See this post on the Onion Community for an interim, manual fix
Status: expect a fix in March
No support for enterprise WiFi that requires a username and password
Status: expect a fix in March
No automatic support for WiFi channels 13 and 14
Status: expect a fix in March

Now I don't know when that page was last updated, the latest firmware is from March 2017, but "expect a fix in March" technically could mean March 2018.

My Omega2+ was apparently delivered with a corrupted/missing firmware, so I haven't been able to test how/if mine works yet, but reading these forums, there are apparently quite a lot of folks with wifi issues (I do wish I'd taken the time to research the Omega2 a bit more before spending the last of my quite modest student budget on one) since early 2017 with basically no input from anyone associated with the project.