One option is to Add URL Rewrite Rules.
Create an API Endpoint in WordPress is a good tutorial as well as The Rewrite API: The Basics.
Monkeyman Rewrite Analyzer is a great way to test what you created. Just be sure to flush your permalinks after each change.
This is also a good example which uses the endpoint to redirect elsewhere.
The biggest trick will be to get your ID for each post on the old site into a form which can be used on the new site.