Learn English – How do we say something started to happen frequently

For (00:45, 01:45, 02:45... 23:45) I would write:

Perform hourly, at 45 minutes past the hour.

where "the hour" refers to the top of the hour (:00).

I think Eliot wanted to match it with the 'do' in the preceding line.

Or it was 'As he does* do' (which — although whimsical — is a grammatical sentence), but he changed the 'does' to 'do' to fit the style and meter of the poem.

^{*(I believe it's the emphatic does but he changed it to do.)}