Package documentation says that on Linux you can use OpenBUGS instead but if this requires installation, you are not going to be able to use it, since you don't have sudo rights on RStudio Cloud.
The classic alternative to Winbugs is jags, but also requires installation.
I suggest you have a look to NIMBLE (https://r-nimble.org/). Depending on your specific BUGS code, it may just work out of the box (and quite faster than Winbugs or jags)
I thought Stan is another alternative/replacement for Bugs and Jags? And from what I remember, there were posts already that showed that you could use it in RStudio Cloud, albeit for rather small models given the 1 GB RAM limit.