You should not take ibuprofen or any other pain medication other than Tylenol without talking first with your primary care doctor
. Ibuprofen, naprosyn, and related medications all belong to a class of medicines called non steroidal anti inflammatory drugs (NSAIDS). One of the effects of NSAIDs is that they impair the function of platelets which, as you know are important for blood clotting. Coumadin, on the other hand, impairs blood clotting in a different way, by impacting the INR level. So, although there is no direct overlap between the two drugs, if you take Coumadin and NSAIDs at the same time, you are attacking the body's ability to do blood clotting in two separate ways. This could greatly increase your risk of having significant bleeding
Unfortunately, there are no other good options for you. Aspirin will also increase your risk of bleeding, so you are stuck taking just Tylenol for pain until you talk with your doctor.
The best option would be to see if you can figure out what is causing the headaches and treat that. For example, if you have sinus congestion, that is something that your doctor will be able to treat independently of using painkillers.