My daughter'* 2000 Lesabre has a whistling noise for about the first 90 seconds of running when first started. The noise is coming from the apparatus just underneath the upper radiator hose. I attached a file with a pic, and it'* the item in the upper right hand corner with the cone sticking from the top....what is this, why does it whistle during initial start up and is there a fix to the issue??

Having a little trouble seeing what it is in the pic, I can only take a guess, and I am guessing that it is the vapor canister.
I'll attach a pic of that part, let me know if this if the part you are referring to.

If it is the vapor canister then it is an EVAP leak, it could be one of the hoses going to, or coming from the canister, inspect them for dry-rot, if they need replacing you can buy the little EVAP line from your local auto-parts store.
If the line looks fine then the canister may be cracked, if so it is not too hard to replace, but it does cost from 93.00 to 100.00
My money is on one of the lines being cracked from dry-rot.

the canister should vent until the purge valve opens and lets the vapors into the engine. shouldnt do that till you are driving though. if anything i would say the purge valve is stuck open. but you should be getting a code for that. you have any codes?

Is he talking about the air pump? On a cold engine it will come on for a predetermined time based on coolant temp.....

What air pump are you referring to Gus?
The only air pump I know of is the compressor for automatic air leveling, that is normally in the rear, underneath the car.

Smog air pump injector. It injects fresh air into the downstream side of the exhaust manifolds. Its to decrease emissions. Worthless IMO because the pumps tend to suck water in and seize.

The whistle is coming from the top of the 'cone' on this device....it lasts for about 1 minute then ceases, until the engine is started again after complete cool down of engine. It'* an irritant but manageable. If it was a vac leak, it'd make the noise all the time. Additionally, there are no MIL'* set.

There is a purge valve inline in the system, during first start up, the PCM conducts a system check of all sensors and devices.

Can you get a audio/video recording and post it to Youtube?

there is a number of parameters that have to be met before the evap does any testing, not sure about the air pump

Can you take a close up picture of the part making the noise(put your finger on it)....