Old boy, like boor in back street, pushes in (8)

'pushes in' is the definition.
'obtrudes' can be an answer for 'pushes' (obtruding is a kind of pushing). I am unsure of the 'in' bit.

'old boy like boor in back street' is the wordplay.
'old boy' becomes 'OB' (abbreviation for a private school 'old boy').
'like boor' becomes 'rude' (I can't explain this - if you can you should give a lot more credence to this answer).
'in' indicates putting letters inside.
'back' says the letters should be written in reverse.
'street' becomes 'st' (abbreviation).
'st' written backwards gives 'ts'.
'rude' going within 'ts' is 'trudes'.
'ob'+'trudes'='OBTRUDES'

Can you help me to learn more?