A sauna has high temperatures but ALSO high humidity. You have to deal with that, too, and protect your electronics accordingly. The DHT22 sensor can handle 100% (or so it says - mine never seem to get over 99.9% for whatever reason).
If you're not interested in the humidity, you may be better off by using a much cheaper, more robust and potentially even more accurate thermistor instead. It just needs a little bit more calculation on the Arduino side.
For the WiFi, indeed a NodeMCU (or WeMOS D1 or other ESP8266 based board) would be a natural fit. They have WiFi built in, and can run reasonably efficient on battery power. Much better efficiency you'll be able to get by using the bare bones ESP8266 processor. Have it sleep, then wake up periodically to read the temperature, send it to the server, and go back to sleep. Be aware that WiFi (and Bluetooth and any other wireless communication) takes quite some power. Your phone could then connect to the server to retrieve the data any time.