Your port of arrival is Bengaluru, and hence the place for immigration check. The customs will thus apply at the port of entry.
And since there would be a custom check, you will have to claim your checked-in luggage, go through the customs, and go to the domestic terminal of Bengaluru airport, visit the counter of Air India and submit your luggage there.
Having just one hour for connection is very little, which leaves no room for any delay. And with Air India, I am sorry to say, delays are likely to happen. A usual immigration line takes about 15-20 minutes for Indian citizens and 5-10 minutes for foreigners. If you don't have anything to declare at the customs, that would just mean passing through the green channel. If you do have however, that is bound to consume way more time. And these apply to both foreign and Indian nationals.
Also note that domestic flight travels in India do not allow any liquids to be carried in hand-luggage. So any of those bottles duty-free perfumes/liquor or the like will need to be stored in your check-in luggage when you do the transfer.
Reference: Mumbai airport customs but the rules are very similar across all international airports in India