Go to the settings, then under networking click the "+ Custom Domain" buttonIt seems like after the trial ends you will have to upgrade plans to keep using a custom domain though.

You can use a custom domain on Railway during the free trial, but once the trial ends you either upgrade to a paid plan or your custom domain stops working.​

How to add a custom domain

• Deploy your app and go to your service → Settings → Networking / Custom Domain.​

• Click + Custom Domain, enter your domain, choose the port (like 8080), and Railway will show DNS records.​

• Copy those into your domain’s DNS (CNAME or A/ALIAS), wait a bit, and it should show “Setup complete.”​
  
  

When the free trial ends

• Free tier = only *.up.railway.app, no custom domains.​

• To keep using your domain on the same account, upgrade to at least the Hobby plan.​

If you don’t want to upgrade

• Make a new Railway account to get a fresh trial.​

• Redeploy your site there, set up the custom domain again, and point your DNS to the new service.