Oh, the original Desire! So can you find a copy of an older app called "HTC Sync"? "HTC Sync Manager" is the current version, but I don't know whether that's compatible with the A8181 (codename "Bravo") - the app that came with that phone was just "HTC Sync".
What data do you need to get off it? Stuff like contacts and messages can be backed-up to the SD card, which can just be removed and put into another phone (or onto a computer if the new phone doesn't have a card slot). You should be able to back up a lot of stuff to Google servers so that they download to the new phone (e.g. when you reinstall apps). You should be able to change the USB mode so that you can copy data without the sync, but that will only let you see stuff on the SD card anyway, so it probably is worth trying the HTC Sync instead (you can always read data from the SD card using a card reader). Without root a backup app like MyBackup (another way of transferring data, backup on one device, restore on another) will be able to back up a lot of stuff but not app data. Unfortunately the non-root backup app that can back up data for many apps, Helium, requires Android 4 because of the way it works, so isn't an option for you.
The other question is, what phone are you replacing it with? Many manufacturers have their own tools for importing data from another phone, e.g. HTC have their own "HTC Transfer Tool" in the Play Store: install that on the old phone (which doesn't have to be a HTC), then you can use that to copy data to a new HTC phone over WiFi. I expect that other manufacturers have something similar.