Do you know a good way to manage cost on AWS for a web app with PostgreSQL?

Generically AWS budgets + budget alerts and cost explorer can help you monitor them and restrict spending. Outside of that, it's just down to your design and how many resources it uses.

Edit: budgets are located under the billing section in the AWS console

What specifically are you concerned about?

The less you use, the smaller your bill.

Something that could help is to not provision a 100GB RDS if you're DB is only 10GB.

You pay for provisioned space not used space.

Grow it as you require.

@SortOfTested I don't know if it's better use on demand or use ec2 instances

@bettoisc
Simplifying my answer:
Use on demand to get started, and then at the inflection point where you have 80-90% usage load on your cluster sustained, go reserved. Reserved instances save a ton of money.

Self-managed EC2 is only really valuable when your application is fully mature, has a lot of sustained load and you need full control/isolation, or have a way to save money vs the RDS solution while accounting for management time overhead.

Tl;Dr Start small with RDS (t3.micro) and just turn the knob up as you need to. Experiments of this kind are cheap, measurable, and easy to back out of.

Highly recommend going serverless. 6 months of development and production costs was about 137$. The majority of that being CodeBuild and other infrastructure or devops stuff. About 0.01 for lambdas. Would have the same postgres costs...unless you go serverless aurora
Architecture is fun!