You could use an .Rprofile file, and source() the file that you want to run when you open the project. E.g. create an .Rprofile file alongside your .Rproj file, and inside that have something like:
source("path/to/your/script.R")
A word of caution would be that doing this risks making your later code/workflow less reproducible: if you rely on that script running then there is a risk things will break for you/other collaborators if that script doesn't get run.