The Metro Marlow in Chinatown fits your budget. Nothing flash. Metro Marlow
The ibis Styles Sydney Central is another budget hotel that is under your limit. ibis Styles.
When I have cruised on Carnival from Australia in the past, the first thing I did if I wanted to do a Carnival shore excursion, was go straight to the excursion desk after boarding. Like you, I was unable to pre book before the cruise.