Start recording when playing a note

Is it possible to start the recording automatically when triggering a MIDI note?

Not by any built in function of Cubase, unfortunately. You can only assign a midi message to a Generic Remote for the discreet command “record”.

It would be very useful to have it as a recording method.

True. Cubase does have Retrospective record though, which for certain workflows can serve as a replacement. If it’s activated in prefs>Record>MIDI, you would play the phrase, then hit Transport>Retrospective MIDI record. The content of the MIDI buffer is turned into a MIDI part on the record-enabled track, and the captured notes end up exactly where you played them in relation to the project.

https://steinberg.help/cubase_pro_artist/v9/en/cubase_nuendo/topics/recording/recording_enabling_retrospective_record_t.html

This works even if you are in the editor.