Exactly! This is the way to use it. Once you have it built as libfortran_stdlib.a for example, you can just carry it over (together with the module and submodule files, .mod and .smod) and don’t worry about compiling it again.
One very easy way to do this is via fpm install as that will put all necessary file in your installation folder (for example, on your pendrive). then you just compile with -I/path/to/stdlib and -lfortran_stdlib.
Let’s not forget this is a Standard Library: long-lasting code with a stable API, not something that’s supposed to change / have to be compiled every day