Some of this is solved by using e.g. systemd, depending on your needs.

> I couldn't have the old and the new version of my service running on the same port...

You can, actually! You just can’t open the port twice by default. So one or both of the processes needs to inherit the port from a parent process, get passed the port over a socket (Unix sockets can transmit file descriptors), or use SO_REUSEADDR.

There are some libraries that abstract this, and some of this is provided by tools like systemd.

Some of this is probably going to have to be done in your application—like, once your new version starts, the old version should stop accepting new connections and finish the requests it has already started.