From Yenikapı, you can get the IDO (Istanbul Deniz Otobusleri) Istanbul Ferry Lines. There are frequent ferries to Mudanya-Güzelyalı (Bursa). To check their schedule, you can visit their website and buy a ticket online: https://www.ido.com.tr/ (Please note that some ferries might be cancelled due to bad weather conditions, in such cases they are announced on the web site as well).
Once you leave the ferry at Güzelyalı Station there are yellow buses to the city center. Take the bus and get off next to the EMEK Metro station. At Emek Metro station you can take Metro to the point where you would like to go in Bursa.
After you exit the airport at the arrivals level you will find a small shop that sells tickets to Bursa (Burulaş). There are yellow buses that depart to Bursa every hour. Take one of them and you will be in Bursa within an hour and half.
The Bus will leave you at the Bursa coach station and you can take either the city bus or a taxi (yellow cab) to the city center. It should cost you around 50-70 Turkish Lira by taxi.
We are looking forward to seeing you in Bursa