The amplifier should be tested at 77 K before putting it in the cryostat, since everything is faster and easier if you can avoid using the cryostat. For this purpose, I have a large, heavy brass block and a Pyrex pie plate. I bolt the amplifier to the top of the brass block, put the brass block in the pie plate, and put the pie plate on a piece of insulating foam. I then put a piece of white plastic over the amplifier.
I put a piece of 20 cm semirigid coax on the end of each cable on the network analyzer before calibrating. These extra lengths of cable protect the precision network analyzer cables from the extreme cold. Set the power on the network analyzer very low, such as -45 dBm, and turn on integrating to get a better signal. Once the analyzer is calibrated, I hook up the amplifier and bias it up. Once I'm satisfied that everything is working, I start filling the pie plate with liquid nitrogen.
The purpose of the brass block is to keep the amplifier at a uniform temperature while cooling, and to slow down the cooling process in general, to reduce thermal stresses. It takes several minutes to get the block down to 77 K. It's obvious when this happens, because the nitrogen suddenly makes contact with the brass in liquid form causing a sudden, loud fizzing sound.
I watch the gain and reflections on the network analyzer while the amplifier cools. The gain will typically steadily increase and flatten out, allowing the bias currents to be steadily turned down.
After the amplifier is cold, I record the full scattering parameters. I then disconnect the network analyzer and connect a spectrum analyzer to the amplifier output. I put a variety of load impedances at the amplifier input (open, short, and 50 Ohms) while checking for oscillations with the spectrum analyzer.
If the scattering parameters look good, and the amplifier doesn't oscillate, then it's ready to test at 4 Kelvin in the cryostat. I don't bother trying to test the noise at 77 K. If you want to know the 77 K noise performance, measure it in the cryostat before transferring helium.
To warm up the amplifier, put it in a small vacuum vessel and pump it down. This is important, or else the amplifier will sweat and become soaking wet.