Of all the department-store food courts in Shinjuku, Takashimaya has the best selection, with three floors of restaurants. Many of the options here are branches of famous chains, like Ningyōchō Imahan (for wagyu, Japanese beef) and Din Tai Feng (for xiaolongbao, Chinese soup dumplings). Most have English menus or plastic food models out front.
Some shops are very popular so you might have to queue for a bit during peak lunch time. The ones on the 14th floor are the priciest.