It really depends on which Generation/year of grill you have. Genny 1-5, 1000 - 5000, Silver/Gold A/B/C through 2004 all had open carts and were built like tanks. My 2004 Genny Gold C has sat uncovered for 12 years and there's not a spot of rust even with all the rain and occasional snow we get here in Ky.
That said, the Weber's I have seen that did have problems with rust were all close cart designs, to include ones like the one you picked up. Most of the parts of the grill will be fine, but the bottom panel of the cart is prone to rusting through.