For the apiserver look at api gateway and lambda - they're really cheap 1M calls on apigateway are $3.50, 1M lambda calls are $0.20
You might want to try AWS Aurora - it's a serverless autoscaling db that you could use in place of postgres
For the basic ui if it's static look at S3 or Netlify
For the always running process you might want to look at Heroku
For the apiserver look at api gateway and lambda - they're really cheap 1M calls on apigateway are $3.50, 1M lambda calls are $0.20
You might want to try AWS Aurora - it's a serverless autoscaling db that you could use in place of postgres
For the basic ui if it's static look at S3 or Netlify
For the always running process you might want to look at Heroku