The simplest method is to use fresh regenerated silica gel inside the cup (half of the cup volume at least). You can maintain a 0% RH for several hours or days, depending on sample permeability and volume.

For the high humidity part of the experiment you can use a saturated salt solution. Depending on salt and temperature you can reach stable conditions in a wide range of humidity (you can easily find tables in the web).

I fear, it will be difficult to get a real 0% RH even using an efficient absorber (you can put it at the bottom of the desiccator, cover it with the perforated plate where you can then put your samples). If technique of Marco proposed above doesn't work for some reasons, you may search for a glove box with a gate and a regeneration system - such systems are quite common. Regenerator will bring you 0% RH (and control it) and 20C you may regulate if needed via conditioning of the room. RT may be sufficient though...