Unit Testing an application is always recommended, but they are no help when it comes to "Black Box" / "End to End" / "Golden" tests.
Having said that, some of the principles of Unit Tests still apply to your task:
- Automate the process - There should be no human involvement besides hitting "run".
Use a dedicated platform for that - Teamcity is one of many. - Isolate all the tests - You should run any subset of them in any order, and get the same results.
If you have a DB, run some script before and after every test, so that each test gets a clean environment (apply to all other statefull modules, like the file system, web cache, etc). - You might as well use a test framework (like NUnit), that already has built in Setup, Test and Teardown entry points, and call your setup script, your API, and cleanup script from there.