You will need a domain name, you can buy one from a registrar such as hover or namecheap (for the love of all that you consider holy do not use godaddy).
You will need a way to expose the server that you set up via port forwarding or similar on your network.
You will need to set up DNS records on the domain you buy to point to your home IP. You may want to figure out a different way to avoid just handing that information out, cloudflare can help with that. You will want to make sure the DNS records get automatically updated if your IP address changes, which is not uncommon for residential ISPs.
You will need to figure out how to get an SSL certificate, Let's Encrypt will issue them for free, cloudflare gives you one if you use them as a reverse proxy.
All of the above is what you probably mean when you say a "url".
I can't help with the synology bit other than to say docker is probably what you will want to use.
Finally, some of this would likely be easier to do on a cloud provider like digitalocean or linode and could be done reasonably cheaply.
Best of luck.