"I am going back to the airport!"
Translation:Visszamegyek a repülőtérre!
So, if you were to say "A repülőtérre visszamegyek" that would put the emphasis on the airport because it is in front of the verb?
Like in "A repülőtérre visszamegyek, nem a házba"
And/Or should this answer be accepted here?
Yes, the emphasis is on the airport in that case but the correct form is 'A repülőtérre megyek vissza, (nem a házba)'
And I don't know what the English sentence emphasizes (I guess nothing)
Pretty much yes, -re (to), -en (on/at) and -ről (from) for repülőtér. This is also true for the root word tér ("square" in the sense of a broad open place) and other compound words that end in tér like csatatér (battlefield) or piactér (open air marketplace).
There are some cases where you might use the -hez / -nél / -től family of endings like if you're going to (the vicinity of) the airport but not really going to the airport proper. Both might be described as "to the airport" in English.
So, why wouldn't it be "Visszamegyem" since "repülőtér" is definite?
repülőtér is not the direct object of the verb. It functions like a prepositional phrase. Only when there is a direct object (with the accusative -t suffix) can the definite conjugation become necessary.