As far as I can tell from reviewing the following documents, it appears to be possible to connect to the Internet. 
 Container services in Amazon Lightsail | Lightsail Documentation
Yes, container services do have public IPs, but they're not stable. When setting up a custom domain, you'll want to point your custom domain to your container service by means of a CNAME record or an ALIAS record. Route53 and Lightsail Domains have special support for ALIAS records and some third-party DNS providers support ALIAS records as well.
Lightsail container services can connect to anything on the public internet that accepts the connection. You may want to check your connection string for typos, make sure that the resource you're trying to connect to is reachable from the public internet, and add logging to your application so that you can get more information about the issue.
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