Function instances should stay alive for a certain amount of time. But you will pay the cold start price one way or another, so naively using Spring Boot is probably not a good idea. Spring Boot Native might help here, but I haven't tried it yet.
But is the web server spun up and kept up somewhere magically? Another poster mentioned something about an ECS task. The reason I am asking is I genuinely don't understand, I thought lambda as an abstraction was all about something which spins up and down when finished.