For your case you are looking to upgrade only CMS, then you can upgrade only CMS and CMS DB and keep the legacy publishing targets set up to follow the SDL documentation for CMS upgrade. not required to upgrade anything on Content Delivery you can continue to use your in-process API and no changes required on your content delivery and presentations.
If you are planning to upgrade Content Delivery then your web app needs to rebuild against 8.5 Content Delivery API.
It's required to involved to scan your existing system, identify and create a report of all items which are deprecated or dropped, it will be based on your web app implementation webform or DXA to get a better understanding to estimate your efforts.
If you have used any custom Deployer Extension – this should work in most cases except you may need to recompile.
Presentation Web App Code – You may require to re-build your code with latest Content Delivery API
APIs – If you are using Broker API, it would work in most cases except few changes, If you use in-process API mostly backward compatible.
If you have used the Web forms, then I would recommend revamping it to DXA and use new microservices architectures to get most of the benefits and performance and capabilities, This ensures the end user is always getting the best out of the technology. It's easy to enhancements new feature in your web app based on DXA releases from SDL community.
I hope it helps.