Assuming that the continent is somewhere on Earth.
First, "rain shadow" has limited applicability so near the equator, because there is very little wind there. Certainly not in the northern part of the continent, which is squarely in the inter-tropical convergence zone.
If the two mountain ranges are comparable to the Himalayas, then (1) they are very very young, and (2) they are hundreds of kilometers across.
Australia has an area of about 7,700,000 km²; the triangular continent being about the size of Australia, this makes the length of each of the three sides about 3,000 km. The continent extends from about 8° northern latitude to about 15° southern latitude.
8,000 meters tall mountains will surely have glaciers on them. Those glaciers will drain both towards the sea and towards the interior. There will be significant rivers draining the interior of the continent; how large will those river be is an authorial choice. If you want them to be lost in the desert like the rivers which drain the Pamirs to the north, so be it.
The two coastal mountain chains being so young, this means that the continent is being actively squeezed from the west and from the east; this will of course raise the land between the mountain chains, so that the southern third of the continent will be a high plateau, comparable with the Tibetan plateau on Earth. Whatever the climate will be there, it won't be what we normally think of as tropical.
There may be extensive lowlands in the northern half of the continent, unless the entire continent is an ancient elevated craton like Africa. If there are lowlands in the northen part of the continent, they will likely be typical equatorial jungle, as they have no reason not to be. Think the north-western part of Borneo.
What the middle part of the continent will look like is anybody's guess. It can be low, it can be hilly. It can be well irigated by the rivers coming down from the lofty mountains. Or it can be a semi-arid savannah if you like. My bet is that it will look like the land around Lake Victoria.