My power steering pump makes a lot of noise early in the morning, especially when it's cold outside. and after it runs for awhile or warm up after a couple of miles, it quietens down. Still works good other than that.

The noise you are hearing is very common,and the fix is easy and cheap. There is a return inlet o-ring on the pump #91345-RDA-A01 that costs less than a dollar, that is letting air get sucked into the system and causing a whine, until it warms up and seals itself. Some Hondas have a recall for this, check if yours does. I also recommend replacing the return hose from the reservoir to the pump at the same time, once it is disturbed, it can cause the same problem. 53731-S9V-A00 $5.28. Get a bottle of Honda power steering fluid at the same time, only use genuine Honda fluid.

Just like Larry said. It's old and tired. It will work for a while. But price one out, price the tools to replace it. You bring it to where I work and the cost will be $400+.

Just getting old. Replacing it is the safest thing to do. Personally I would not worry until it leaked or got worse, but that is me. Replacement is always the safest thing to do.