I'll take a stab at this. I'm not a climatologist, and it's hard to make an educated guess because there's nothing at this scale ringed by mountains on earth. How did these mountains come to be? What kind of tectonic activity could create this type of landmass?
These will be generalizations, because obviously many climates exist within a three thousand mile radius.
At Sea: The Doldrums. Around the Equator there's an area of low wind, where it's very humid and the air just sits. Around the equator is the intertropical convergence zone where airflow is vertical rather than horizontal. The landmass will cause uneven heating, which will create some alternating breezes depending on the time of day, but the overall wind from the ocean will be negligible.
On the continent, winds will be driven by albedo differences combined with the Coriolis effect since there are no oceans or waterbodies. The continent is very large, so within the mountain ranges, there will be winds, though they won't have time to pick up speed like they do over the ocean. I would expect a clockwise gyre inside the mountains, sucking up a bit of moisture from the openings.
At higher latitudes, the continent's shores and mountains will be forested and very moist. It will rain all the time. High winds will be the rule, with frequent thunderstorms. These storms will spawn hurricanes on the landmasses on the other side of the oceans.
Depending on the placement of the gaps, they could profoundly impact the continent's weather. If they're all at the equator, they won't do very much due to the low winds, but if placed in higher latitudes, the low pressure areas created by rising air over the land will divert the moisture laden winds inward.
Outer Ring: Dry beaches at the tropics give way to scrub and eventually forested regions at the higher latitudes. Elevated areas go from very dry at the equator to alpine in the high latitudes.
The inner portion of the continent will be dry. Not desert dry, because the continent will presumably have springs and some moisture will be sucked into the gyre from the gaps. But, like ratchet freak suggested, probably savanna, like the southern part of Africa (which does have some elevation separating it from the ocean).