Well...I grew up in the East End of London and although time has dimmed the brain cells... here goes!
A monkey is £500.
Another meaning in CRS for monkey is "wench" (monkey wrench) which makes much more sense to me in the context of this rhyme. Especially as later in the song we see the "stick" (= well... obvious!) "knocking off" (having sex with) the monkey.
Since the rhyme is about being broke, the various activities of the perpetrator cause him to have to "pop his weasel" (i.e. pawn his coat (weasel and stoat)).
Treacle = treacle tart (sweetheart) and I would hazard a guess that "tuppenny rice" = dice (= gambling) i.e. another way of losing money.
The City Road is a real street in London and The Eagle was a famous pub as described by others above.
Hope this helps a bit.
All the best