Yeah, opposite rotation would make sense, and without latitudinal lines its not necessarily easy to discern details… however, if it isn't backwards, then I'm not sure how it greens up west of the desert with earthly equatorial storm movement. But I would have to see your currents, air, etc detailed to have a better idea. Earthly flow, that area might be a good candidate for monsoons.

Also with backwards flow in currents that big eastern desert would have a cold water shore, rather than warm water, which would assist in desertification. However, it might be northerly enough to experience weather traveling east to west (in backwards spin) which would bring moisture from the ITCZ (which would rise high with such a large land mass keeping the atmosphere warm, theoretically) which could hurt the desert scenario. With an earthly flow, the desert makes a good bit of sense, as a rain shadow, but still, the desertification of that peninsula seems highly unlikely without justification. And of course, the temps of the world in general can change everything. I'd also look into the plate tectonics if you want to go nuts, LOL.

The scale of this continent also throws a wrench into the hypotheticals.

Work on currents, winds, ITCZ and seasons, plate tectonics… No geography instructor or anyone on the internet will really be able to give hard answers to the questions, only suggest possibilities and probabilities. Right now I would call it possible, with the right explanations, probable? Not sure.