For sure it is possible You can reach through car/train and road Mumbai to Delhi/Chandigarh flight From delhi you can many buses ( too long journey ) You can get bus from Shimla/Chandigarh/Dharamshala
The best you can do , to have a flight to Delhi and then from Delhi, there is a bus stand name "Majnu ka Tila", you can reach here metro and nearest metro station is "Vidhan Sabha" and bus will take approximate 12-13 hours based upon stoppage on the way. Happy to help if you need further assistance.
5 Nights 6 Days Mumbai to Manali itinerary by Train
Day 01 :: Get train from Mumbai to Delhi/Chandigarh.
Day 02 :: Get a volvo for Manali.
Day 03 :: A car will pick you from Manali volvo stop and drop you at the Hotel/Resort in Manali. Check in Hotel/Resort and go for Tour rest.
Day 04 :: Full Day Rohtang Pass tour by Indica/Alto Car.
Day 05 :: Check out to the Hotel/Resort in Manali and go for Half Day Kullu Manikaran tour by Indica/Alto Car and drop you at the Manali Volvo Stand.
Day 06 :: Reach at Delhi/Chandigarh and Get train for Mumbai from Delhi/Chandigarh.