the airpump..pumps addtional air into the fuel system during start up when the motor is cold. What you hear is the cheap bearings in it. It sits ontop of the engine on the driver's rear side. It has nothing to do with the transmission.
It can be a costly part, but it is a part that most can install themselves. But, I have 155,000 on my allroad and the pump has been noisy for most if those miles.

allroad FAQ SITE.
2003 Allroad 6 speed manual:
DRIVETRAIN:
• APR 5.0 93 Octane
• Stratmosphere Hyperboost Diverter Valves
• Toy Guy Short Shifter
• AWE-Drive Train Stabilizer.
• HP-2 Caliper Reverse Mod
• ECS Slotted F & R Rotors
• PBR Metal Master Pads F&R
SUSPENSION:
• Hotchkiss H-sport stabilizer bars F & R
• S4 Heavy Duty Tie Rod Ends Inner and Outer
• Falken Ziex 912 UHP-A/S
• Borbet FS 17 x 8 Wheels
• HR 12MM Spacers F&R
• 402 mod down 20MM
INFOTAINMENT
• Kuda Cell Phone Mount w/hardwired DC plug.
• Parrot Blue Tooth Cell Phone/GPS Module
• Palm Treo 700 with Tom-Tom Navigation
• Ice+link Plus w/Mini Ipod
• Sirius Replay Satellite Radio
• Hardwired V-1
MISC:
• Audi Trailer Hitch