I do not know if it will be enough ventlation howver you can install a vent in the closet door that will allow air to circulate better.. I did that in my rig 10years ago and to date no problems.
Why is it better to install upright instad of on it's side:
There is a device called a heat sink.. it is basically a hunk of metal often with fins that "Sinks" (Draws) the heat from something..Also transformers generate heat...UPRIGHT is the most efficient cooling of the components..Sideways the cooling is not as efficient. Mine is upright (only choic where it sits)