NonEEA Spouse Permit in Uk

Hiya, I'm a USA citizen who has been studying in the UK and will be marrying my Irish boyfriend at the end of the month. We've been living in the UK since fall of 2017, with him working full-time and me studying. I'd like to apply for the Non-EEA spouse permit after we're married since my student visa will be ending shortly.
My question is this, I'm aware the application needs to be made outside of the UK, but what exactly are the limitations on this? Is it possible to file the application from Ireland and then return to the UK as a visitor while we wait for it to go through? It is our understanding that the it can take up to six months to go through. Any advice or suggestions?

Once married, your rights are automatic and you can apply for an eea residence card within the UK. No need to apply from abroad.

Thank you so much for your reply! The UK.gov site is fairly helpful, but not always clear for different circumstances. I appreciate your help.