Can you take 1000mg of naproxen and 1800mg of Tylenol together at the same time?

You can safely take Tylenol and naproxen at the same time since they are different types of medicine. For example naproxen is an NSAID (non steroidal anti inflammatory drug) and Advil (ibuprofen) is also an NSAID. It is recommended to not take naproxen and ibuprofen together because it greatly increases the risk of side effects. With Tylenol (acetaminophen) you need to make sure you aren't taking anything else with acetaminophen as an ingredient especially if you are taking this large of a dose.
These are large doses for either of these medicines. I take prescription strength naproxen and it's 500mg twice a day. Taking large doses of NSAID's increases your chances of stomach issues and bleeding and large doses of acetaminophen increase your chances of liver problems. I would suggest calling your doctor and discuss what to take with him/her.