Is AWS Fargate included in their "first year for free" plan?
Solution 1:
Fargate is not on the free tier list. I suggest you use ECS with a single EC2 instance in the ECS cluster, which will work in a similar way.
Why would it matter if your EC2 instance has low utilization? That's ideal for performance. You can drive slow in a fast car!
Alternately you could also use API Gateway in front of lambda serverless compute. That will probably be cheaper long term. Lambda supports Python.