Converted `api-routes-rest` example to TypeScript to match Contribution docs.
## Documentation / Examples
- [X] Make sure the linting passes by running `pnpm lint`
- [X] The examples guidelines are followed from [our contributing doc](https://github.com/vercel/next.js/blob/canary/contributing.md#adding-examples)
Co-authored-by: Balázs Orbán <18369201+balazsorban44@users.noreply.github.com>
