How should I approach upgrading the version of Elasticsearch? Can operator handle just changing the configuration from version: 7.2.0 -> 7.3.0 and kubectl apply?
Absolutely! That's what the operator is for.
Just reapply the same yaml specification with a different version. ECK will automatically handle creating the new nodes with the newer version, migrating data from old nodes to newer ones, then removing older nodes.
Starting the next release of ECK (1.0.0-beta1), ECK will do it in a rolling upgrade fashion (taking down one node at a time, while preserving the PersistentVolume for the upgraded node), making the whole operation faster.
In the current version (0.9), data will be migrated away to newer nodes as described above. No downtime should be expected on any index.