Landing at ORLY

CC,

I'd suggest using the OrlyVal train from ORY to Antony station, and then connect with the RER "B". The closest RER station appears to be Châtelet Les Halles, so you may have to transfer to Metro or walk from there. It would help to know exactly where your apartment is located.

Ken - the apartment is not far from the RER station in les Halles, I could probably walk. Thanks!

The OrlyVal/RER B route Ken suggests is fastest, but the OrlyVal is pricy. If you want to save some money, take the airport shuttle bus to the Pont de Rungis-Aeroport d'Orly RER station and the RER C from there to Chaletet-Les Halles. For a zoomable map of the Metro/RER system, go here: http://tinyurl.com/6xmvms4

Airport websites are the first and best place for information on transport. http://www.aeroportsdeparis.fr/ADP/en-GB/Passagers/Home/
I agree the Orly automated shuttle is costly, by my estimate about two euros per minute.

CC,

One other point to mention. If you're planning to buy a Paris Museum Pass for your visit, it's very easy to do that at ORY before heading to whichever transportation you prefer to use. Check their website for details on the Pass.

When choosing transportation, I usually prefer the most direct, easiest and quickest method when hauling cumbersome luggage, regardless of cost (within reason of course).

Good recommendations everyone, thanks!

There are other options which are bus / rail combinations. Either Orlybus to Denfert-Rochereau then RER B, or Air France Cars (coach) to Gare Montparnasse, then metro line 4.