Theoretically you will get the same shutter speed with both cameras. In practice it will vary slightly, due to differences in metering sensors and algorithms, as well as differences in actual f-stop, f/5.6 is never exactly f/5.6. Even if you use the same lens, you will have to zoom (otherwise you're getting a different picture with different content, thus the comparison doesn't make sense) which can change the actual f-number slightly even though the camera still says f/5.6.

Whether you get the same image brightness is a separate question, but again theoretically you will but in practice it can vary depending on the camera's actual (as opposed to reported) ISO sensitivity and RAW processing algorithm (and image profile settings). Within the same manufacturer the difference in sensitivity is probably not enough to worry about, however between brands it can be more (Canon used to be 1/3 of a stop more sensitive than Nikon).

Theoretically you will get less noise with the D800 - its larger sensor will collect 2.25 times as much light from the f/5.6 lens. More light means more photons (averaging out the photon noise) and a larger signal (which helps marginalise the electrical noise from the sensor). In practice there are many other factors which differ between sensors, pixel count, quantum efficiency, read noise, many of which go against the larger sensor. However the extra light collected is a significant advantage.