To get from Istanbul to Ephesus by bus, you should take a bus from Istanbul to Selcuk, which is in walking distance of the famous ruins in Turkey.
Buses run daily from Istanbul to Selcuk, and the journey takes around 10 hours. You can either take a daytime service or an overnight one. The latter is a good way to save money on a hotel or hostel if you are traveling on a budget.
Several companies run buses on this route, including Kamil Koc and Pamukkale. Timetables and prices change frequently, so make sure to check when you arrive. There is usually no need to book in advance unless you are visiting Turkey during a public holiday period.
If you don’t want to take a bus from Istanbul to Ephesus, you can take a charter flight instead. Some tour companies allow you to make a day trip from Istanbul to Ephesus if you fly between the two.