As your objective is to reflect the light, rather than explicitly paint the shade (more impractical), consider the following.
One can purchase silver reflective mylar sheet from various sources (Amazon) at various prices. A 4 foot x 50 foot roll is approximately US$30 with some variation therein. Even an "emergency blanket" made of mylar would serve, although removing the wrinkles may be problematic.
The sheet (or blanket) could be attached on the inside of the roll up using adhesive tape. Mylar is quite thin and would have little impact on the roll thickness.
Note that such an implementation would reflect the sunlight back toward the glass and is likely to heat the air between the shade and the glass. Convection heating will result. It may be less than that experienced by the bare shade, however.
One can apply self-adhesive reflective window film to accomplish useful reflection, but as this is an apartment, such semi-permanent modifications are often discouraged.
Amazon listed item is source of image, currently out of stock but representative of other similar products suitable for this application.