Is it necessary to migrate to .Net Core?
Yes, as Lambda runs on Linux.
If it is, can anyone tell me how much development that requires considering my set up?
You will need to pay specific attention to the dependencies to external libraries, as they need to support .NET core and might add a substantial load time to your lambda function.
If you have an existing .NET API developed in 4.5, I would strongly advise against trying to migrate the whole API. Lambda's pay-for-what-you-use model is very attractive, but largely incompatible with ASP.NET MVC/WebApi's server model.