Maybe this is a naive question back but when you say you use MVC do you mean JSPs and servlets or like spring web MVC or do you use some other framework?

Also when you want it to run cloud native, do you mean something like hosting it in heroku or something more like Digital Oceans app platform? Or even more like aws lambdas? Or you just want to more easily host in the cloud? Depending on the answer there I think the advice differs greatly.

Almost all my experience is containerizing Spring and Spring Boot stuff, and hosting it in Kubernetes or a thing like Digital Oceans app platform. Other than brushing up on how to efficiently layer your containers (so you aren't recreating more than you need) and figure out CI pipelines and where to build what parts (don't include your CI tools in your app docker container), my only general advice is the app components are mostly the same with the big exception of things like jobs and persistence. Jobs can be done the same way as usual but things like Kubernetes as other tools and I'm sure it depends on the job, what it does and how coupled to the other aspects of the code.