I'm very curious as to how Asa might go about the role, assuming he will indeed be cast. It'd definitely be considerably different from how Tom Holland or pretty much every other actor I've seen recommended would go about the role.
Asa would definitely knock out the quieter, more intense moments that we're likely to get via Peter dealing with some amount of teenage angst (I doubt they'd go too far with that). He might channel out some of his performance from X+Y by portraying Peter as being a bit more quiet and insecure and thus use Spider-Man as a means for him to come out of his anti-social shell.
I also think he'd be an absolute perfect fit for Peter and Spidey if they do end up adapting the Symbiote Arc over the course of the sequel solo films, as I suspect they will. If nothing else, I think most people would agree that Asa can definitely nail the less common, darker side to Peter.
As for how Asa might handle Peter/Spidey's more comedic elements, I think he might go for a more sarcastic or snarky approach. He's got a knack for that dry British comedy in real life, as seen in his interview a few years back on Conan. Going back to my previous post in comparing the casting of Michael Keaton as Batman to Asa as Spidey, just because we haven't seen Asa do very much humor before, doesn't necessarily mean he can't do it.