Using a server charging me by the hour

It is possible to set it up using the methods in the following documents.
https://lightsail.aws.amazon.com/ls/docs/en_us/articles/amazon-lightsail-tutorial-launching-and-configuring-wordpress

In addition, the following document describes lightsail's approach to fees.
https://aws.amazon.com/lightsail/faq/?nc1=h_ls

Lightsail plans are billed on an on-demand hourly rate, so you pay only for what you use. For every Lightsail plan you use, we charge you the fixed hourly price, up to the maximum monthly plan cost. The least expensive Lightsail plan starts at $0.0047 USD/hour ($3.50 USD/month). Lightsail plans that include a Windows Server license start at $0.01075 USD/hour ($8 USD/month).

EC2 instances are all based on hourly rates. As you have identified, Lightsail instances package instance and bandwidth into a flat monthly charge. There is not a migration path from EC2 to Lightsail. You will need to create a Lightsail instance and then install your software and move data to the instance manually. Once you are happy with the Lightsail instance you can terminate the EC2 instance.

Note: Always a good idea to snapshot the instance prior to terminating it just in case you miss something and need to get back to it.

Hope this helps you!