First, RStudio does not ship with R and use any of your R version you have installed and tells it to use. By default, it is your default R version on your system.
It seems you have Microsoft R client installed for RStudio to use.
You can change that by installing a new R version for CRAN and pointing RStudio to use it.
Changing it this way is the correct way. You can even point to a non MRAN repo like
options(repos = c(CRAN = "https://cran.rstudio.com"))
Then your default repos will be this new one you change it too so yes, running update.packages will update the package.
You can install only the package you want to update by reinstalling them one by one.
to update a subset of packages, I think this is like that
update.packages(oldPkgs = c("bookdown", "rmarkdown"))
as explained in ?update.packages
What did you try after changing the repos to a new MRAN snapshot ?
What is the issue ?