Nitpick, but there is no Serverless for RDS, only Aurora. The two are wildly different in their architecture and performance characteristics. Then there's RDS Multi-AZ Cluster, which is about as confusingly named as they could manage, but I digress.
Let's take your stated Minimum ACU of 1 as an example. That gives you 2 GiB of RAM, with "CPU and networking similar to what is available in provisioned Aurora instances." Since I can't find anything more specific, I'll compare it to a `t4g.small`, which has 2 vCPU (since it's ARM, it's actual cores, not threads), and 0.128 / 5.0 Gbps [0] baseline/burst network bandwidth, which is 8 / 625 MBps. That burst is best-effort, and also only lasts for 5 – 60 minutes [1] "depending on instance size." Since this is tiny, I'm going to assume the low end of that scale. Also, since this is Aurora, we have to account for both [2] client <--> DB and DB-compute (each node, if more than one) <--> DB-storage bandwidth. Aurora Serverless v2 is $0.12/hour, or $87.60/month, plus storage, bandwidth, and I/O costs.
So we have a Postgres-compatible DB with 2 CPUs, 2 GiB of RAM, and 64 Mbps of baseline network bandwidth that's shared between application queries and the cluster volume. Since Aurora doesn't use the OS page cache, its `shared_buffers` will be set to ~75% of RAM, or 1.5 GiB. Memory will also be consumed by the various processes, like the WAL writer, background writer, auto-vacuum daemon, and of course, each connection spawns a process. For the latter reason, unless you're operating at toy scale (single-digit connections at any given time), you need some kind of connection pooler with Postgres. Keeping in the spirit of letting AWS do everything, they have RDS Proxy, which despite the name, also works with Aurora. That's $0.015/ACU-hour, with a minimum 8 ACUs for Aurora Serverless, or $87.60/month.
Now, you could of course just let Aurora scale up in response to network utilization, and skip RDS Proxy. You'll eventually bottleneck / it won't make any financial sense, but you could. I have no idea how to model that pricing, since it depends on so many factors.
I went on about network bandwidth so much because it catches people by surprise, especially with Aurora, and doubly so with Postgres for many services. The reason is its WAL amplification from full page writes [3]. If you have a UUIDv4 (or anything else non-k-sortable) PK, the B+tree is getting thrashed constantly, leading to slower performance on reads and writes. Aurora doesn't suffer from the full page writes problem (that's still worth reading about and understanding), but it does still have the same problems with index thrashing, and it also has the same issues as Postgres with Heap-Only Tuple updates [4]. Unless you've carefully designed your schema around this, it's going to impact you, and you'll have more network traffic than you expected. Add to that dev's love of chucking everything into JSON[B] columns, and the tuples are going to be quite large.
Anyway, I threw together an estimate [5] with just Aurora (1 ACU, no RDS Proxy, modest I/O), 2x ALBs with an absurdly low consumption, and 2x ECS tasks. It came out to $232.52/month.
[0]: https://docs.aws.amazon.com/ec2/latest/instancetypes/gp.html...
[1]: https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/ec2-inst...
[2]: https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide...
[3]: https://www.rockdata.net/tutorial/tune-full-page-writes/
[4]: https://www.postgresql.org/docs/current/storage-hot.html
[5]: https://calculator.aws/#/estimate?id=8972061e6386602efdc2844...