Videos, photos and information on the reconstruction of a 9th-century Arab ship, and its historic voyage from Oman to Singapore.
Assuming the course to be from Oman (say northern Oman, Muscat) to Sri Lanka, June to September are not particularly good months. The SW Monsoon starts in June and generates strong winds and heavy seas. Many ports on the Indian West coast are closed, and I would assume a similar situation arises in Sri Lanka.
October is the transition period between the SW and NE Monsoon. The wind is lighter, variable. The best time would be from November through April, when the NE Monsoon provides a fairly reliable wind which is gentler than the SW Monsoon.
The Jewel of Muscat sailed from Oman on 16 February, mainly because she must sail a passage all the way to South East Asia, using the NE Monsoon to cross to India and Sri Lanka, then picking up the SW Monsoon to sail to South East Asia.
The ancient navigators recommend embarking from Oman between mid-February and the 7th of March, and in turn leaving Malabar in India on April 16/17th for South East Asia. However, whether that leaving date would include a stop in Sri Lanka en route is not addressed.
Some of the best information comes from G. R. Tibbetts’ translations of the Arab navigator’s texts of about 500 year ago (Tibbetts, G.R., 1981 - Arab navigation in the Indian Ocean before the coming of the Portuguese, Oriental Translation Fund, New Series 42. Royal Asiatic Society of Great Britain and Northern Ireland). This is available in some university libraries and at Amazon, Alibris, or other rare book sellers. Another interesting reference is James Horsburgh (Horsburgh, J., 1836 - India directory, or, directions for sailing to and from the East Indies, China, New Holland, Cape of Good Hope, Brazil, and the interjacent ports: compiled chiefly from original journals at the East India House, and from observations and remarks made during twenty-one years experience navigating in those seas. 4th edition, London: Allen and Co). Unfortunately this is a rare book that is only found in rare book collections of some libraries, but has a wealth of information.
There are also a number of internet sites describing the weather patterns over the Indian Ocean so a quick internet search should give you valuable clues.