Yes, if viewed at the same size, but at 100% they will be identical. The 1/focal length rule was designed for 35mm cameras. You want to use the effective full frame focal length. However it is only a rule of thumb and the pixel density of the sensor also has an effect. The ability to see camera shake is much more pronounced in a higher resolution camera than a lower resolution one. In your example, if you crop the photo you keep the same pixel density, so when viewed at 100% they will be identical.
For example an iPhone has a focal length of 4mm, which is equivalent to a 30mm focal length on a full frame camera. So you would want to shoot at at least 1/30s not 1/4s.
This is where the pixels per inch come into effect: Higher resolution camera will depict the given PPI. So for example, a 5DMK2 has 21mp, which is equivalent to a 8MP APSC sensor. What this means is that you take an image of a 5D MK2 using, let's say, a 35mm lens, and do the same thing with a 8MP APS-C (30D, 20D) you'll get the same exact image, when you take the FF camera and crop it to APS-C size in POST. Same "camera shake" as the 5D MK2.
Therefore, it's not the size of the sensor that should depict the general rule, but more about the pixel density of the sensor!
D800 (36mp) is about the same as their D5100 (16mp) so basically, you can take an image from the D800, Crop it 40% less, and you'd get the SAME EXACT image as the D5100, if you'd use the same lens. Same camera shake, same everything. They're both the same pixel pitch, so essentially, they literally chopped 40% of the D800 sensor and gave it to the D5100 APS-C camera.