This is a fantastic question. Do you mind clarifying how RStudio is being used? Is this RStudio Server Pro?
If so, the general recommendation is not to upgrade R (for the problematic reasons of installing packages / breaking content / etc. as @mara mentioned). Rather, we recommend installing multiple versions of R side-by-side, and RStudio Server Pro allows users to decide which version of R they want to use for a given project.
For instance, a legacy project may keep running on R 3.4.2, where a new project may start on R 3.5.0 . The other option, if you do want everyone to migrate to R 3.5.0, is giving them the ability to slowly transition their work over a specified time-frame (rather than everything breaking on upgrade) before removing the old version of R. More on these varied approaches is written here.
There is more about this paradigm in the following RViews article:
Also, although it is for a different professional product, the RStudio Connect Administrator guide has some additional directions on how to build R from source, which enables having multiple versions installed side-by-side.
If you are not using RStudio Server Pro, you can still follow the build-from-source approach. However, only a single version of R can be used with RStudio at a given time. As a result, there will likely be a good bit more expedience necessary in migrating content immediately to the new version.
If you are using RStudio Server Pro, then you should have a selector like the following that allows you to select which version of R you want to use for a given project. It is possible for the admin to set defaults and specify which versions are available:
For those more interested in why R packages are generally not portable across different versions of R, this RPubs article is an interesting read.