1U is referring to the height of the rackmount case. 1U is shorter than 2U which is shorter than 4U, et cetera. A heatsink designed for a 1U case would be low-profile, in which case if it can properly be mounted to your motherboard it may be what you are looking for.
I have a few cases that have a side mounted PSU, and sometimes it is a pain finding a heatsink that will adequately cool the CPU yet still fit properly inside the case. For a while I had a volcano 6 with a slimmer fan (A coolermaster fan that was about a third of the height of the v6 fan) cooling an overclocked thunderbird... my lord was it an inferno in there, average CPU core temp hovered around 75 C.