While I think it highly plausible that a teacher at Hogwarts could be as young as 18, I can't see young teddy becoming a teacher at Hogwarts - though it's possible.
The problem is not the age of the teacher when they start, but rather when they leave. It seems to me that wiazrds and witches can remain in their careers far, far longer than their muggle counterparts. A 100 year career seems fairly common, and we have evidence of people like Dumbledore and the exam marking witch going on for far longer. Thus we don't tend to see more than three Professors for a subject each century - barring Defence Against the Dark Arts.
Given that Professor McGonagall became headmaster in the year of Teddy's birth, the Transfiguration teacher would only have been in the job for 20 or so years in your story. Given that the only Hogwarts teacher we have seen retire was Professor Kettleburn - and then through loss of limbs rather than age - it seems unlikely that the post would come to be open. Professor McGonagall herself served in the position for around 50 years (the reference is in OotP, go find it yourself if you care that much. ) So unless the new teacher was someone of similar age, then it's very likely they'd still be teacher when Teddy was in his 60's.
Of course, rules - like pie crusts - were meant to be broken. It seems unlikely that the Transfiguration teacher would come to an untoward end, but it's possible. If you are desperate to put Teddy in the position, then I'm sure you can come up with a reason.
I remain mystified, however, as to why anyone would want to write about Teddy in the future, since we know absolutely nothing about the character; he was brought up by people we didn't know, and given Tonk's metamorphagis we don't even know what he looks like. At that distance from the canon world, canon seems a pretty distant memory anyway. Still, your call on all counts.