I just purchased a used (but looks virtually new) Agilent U1602A on eBay as well. This unit was built in Korea around mid 2007 and has the original calibration certificate. I'm noticing some things and wanted to compare notes with those of you out there (including the original poster Zaoka) who own this model. I'd also be curious for any owners of the Agilent U1602B (or U1604A or U1604B) to chime in so I can assess whether my unit is working correctly or not.
Observations:
0) SELF CAL. It passes the internal Self-Cal but there is about a 1mV DC offset on the Scope inputs. One is negative, the other is positive. Is that typical?
1) BATTERY LOSING CHARGE. The battery only holds a charge for about 20 minutes, but should be 4 hours. The indicator shows it's fully charged, but it then drains rapidly. It's about $80 for a replacement battery from Agilent -- is there any hope that this can be restored? Supposedly the meter wasn't used much (it still has the original LCD plastic protector and original packaging) and it does look virtually new. However, the 7.2V Battery is probably suffering from low usage and extended storage. Will multiple charge/discharge cycles help to restore the life? The manual mentions for very first time use to charge it with the unit ON for 25 hours. I haven't tried that yet. I did the quick charge method overnight with the unit turned off.
2) STRANGE VOLTAGE RAMP IN DCV METER MODE. When set to the Meter mode and DCV is enabled, I see a regular ramping in voltage from about 100mV to 650mV (approximately), which repeats on a regular basis. This is with nothing connected to the Meter inputs. I also get the same result on the logging mode -- and see a repeating sawtooth on V vs. T. Is this some kind of demo or test mode? What is going on here? If I short the inputs together, then it reads 0.0 mV, as expected. But what is this voltage ramping thing? Do all U1602A / U1602B / U1604A / U1604B units exhibit this behavior? I didn't see anything in the manual about it, so was curious whether this is some kind of auto-ranging trying to kick in or something...while it's looking for an input signal.
3) CAN'T SET DATE. It is not possible to set the DATE or TIME on the unit. I did notice a HEX code where the time is and read an Agilent service note that this means the internal battery backup (CR1220) is dead and how to replace it. So, that is likely the cause of this issue. Just pointing it out for those who may eventually experience the same issue. Do the new "B" models eliminate the battery or use a different battery? Just curious. And how long should this coin-cell battery last under normal use for this unit?
4) CAN'T RESET FACTORY DEFAULTS. Not sure if this is related to the coin cell battery issue above, but it is not possible to restore factory defaults
5) LOGGER MODE DOESN'T RESCALE CORRECTLY. When logging in the logger mode, scaling doesn't seem to work right. For example, if I have a lineraly increasing resistance -- say starting at 1K going to 10K in 1K steps....it will show an expected "staircase" pattern. However, when it reaches the overrange for that display setting, the scale will adjust to accommodate the higher value, but the historical part of the graph doesn't scale accordingly. As a result, all historical data on the screen is not scaling in proportion to the range change that took place. This happens both on the Y axis, as well as X (time) axis. When it reaches the end of the time axis and rescales the time, it doesn't adjust the historical part of the graph to shrink as well. This is a major bug and I don't know whether this could also be due to the memory battery issue (even when plugged into AC) -- or whether it is really a software defect. Can someone verify the performance of their unit to see what should happen? Please also state the firmware version.
6) DMM METER MODE SLOOOOOOOOOW. This issue was noted in the post above. It is almost useless -- as it takes several measurements to stabilize over several seconds. And even then, it seems to kind of jump around more than I would expect. Is the processor in the "B" model faster? Anyone know?
7) LCD DISPLAY FLICKER. There is noticeable flicker on the LCD display (especially against the gray background). I tested it without lights on to make sure it wasn't interference between the 60Hz and the display raster. It even flickers outside. It's not horrible -- but you can see it "twinkling" with some subtle random background noise which seems to be raster/scan related. Does anyone else see this issue on the "A" model or the "B" model? Is the display in the "B" model any better than the "A" model?

FIRMWARE. There may be a more current firmware version that could address some of these questions and I'll try that next -- but the release notes don't really say much about these particular issues. For reference, the current loaded version is 2.5.6 with Key Version 3.2.1. What is the Key version?
Hopefully someone out there sees this and we can compare so I understand what's expected and what needs to be fixed...
Happy testing!