Are there some recommended approaches for running Bitcoin Core on a laptop with constrained storage capacity?
1. Use pruned mode.
2. Replace current storage drive with new internal storage drive to the laptop, which have bigger capacity.
Is it possible to use an external hard drive as the data directory (datadir)?
It is possible.
How does this setup work, and would I need to pay attention to?
You just need to specific where should Bitcoin Core store it's data. Just be careful sudden power loss or disconnect on external storage could cause corruption, which force you to perform reindex and even redownload corrupt blockchain data.