Flybe fly from Brussels to Southampton on Monday, Tuesday and Thursday mornings and every evening except Saturday. Prices are reasonable but bags are extra. The luggage allowance is 20 Kgs. Alternately you can fly from Brussels to Heathrow and get a bus direct from the airport. (National Express) British Airways, bmi and Brussels Airlines fly the route and the first two have a sale on at the moment so the prices are good. With BA at least the luggage allowance is 20Kgs but you can pay for extra up to about 32 Kgs. The bus will set you back about £15.
There seem to be two options,neither of which is easy. Flybe fly direct to Southampton but there are morning flights only on certain days. Website is flybe.com. Airport is about 5-7miles from port. You can also go by train - Eurostar from Brussels to London St Pancras, which takes around two hours. eurostar.com. Then you have to get from St Pancras to Waterloo by taxi (20-30mins) or subway and train from Waterloo to Southampton which takes around 1.5hours but trains run about every half hour. nationalrail.co.uk. Southampton station is one mile from dock. Probably train is best option and my guess it would take around 6 hours overall. Brussels is one hour ahead of UK so leaving at 8am would get you to Southampton around 1pm. Leave an hour earlier to give you some contingency.