Yes you can visit Tawang In June. Tawang is Prime Destination of Arunachal Pradesh. You should reach Gauwhati from Pune by Air Or Train.
Start your Tawang tour from Gauwhati.
Day -1 Gauwhati to Dirang Night Stay
Day -2 Dirang to Tawang
Day-3 Tawang Sightseeing
Day-4 Places around Tawang like Bumla Pass, Madhuri lake
Day-5 Tawang to Bomdila
Day-6 Bomdila to Gauwhati