Yes, the weather is mild both in Beijing and Shanghai during Oct. It is suggested that you stay in Beijing for 3~5days and then Shanghai 2~3 days. One week should be ok for visiting these two cities. Transportation between Shanghai and Beijing, you could choose the night train or quick flight. The real-time infor could be search at: https://www.travelchinaguide.com/china-trains/
https://www.travelchinaguide.com/china-flights/

For accommodation, you could consider these chain hostels which cost about 200RMB/night.