If Vishnu is eternally youthful and Advaita Acharya is Vishnu...

Then why does he appear old with white hair and a white beard?

The same way baby Krishna appears to pass stool when in fact he has no inner digestive system (being sat-chit-ananda rupa).

Advaita acarya didn't have a beard...

Yes he did.

Srila Prabhupada Bhaktisiddhanta Saraswati Thakura clearly said that Adwaita Acarya Prabhu never had a beard and used to keep clean shaven. But the bauls (one of the thirteen apasampradayas) wanted to make him a baul and put a beard on his face. On the Deities at the Yoga-Pitha or Srivas Angan as well as any other Gaudiya Math temple you will never find a beard on Adwaita Acarya's face. But at the Mayapura ISKCON temple as well as other temples (in San Diego and Hawaii) we can see Adwaita Acarya with a beard. Are you following the bauls?

Please cite your evidence, otherwise don't bother posting it. Simply you saying Bhaktisiddhanta said this and that is meaningless. By the way, you just copied and pasted a segment of a letter from Bhagavat Maharaja word for word.

Chaitanya Bhagavata Madhya Lila, 16.99 (If someone has a copy, they can check the bengali and see what it says exactly):

"Placing His hand on His beard and knitting His eyebrows, Advaita Acarya, the master of Shantipura, loudly roared. Then He began to dance."

It appears Narayana Maharaja also accepts He had a beard:

After some time, however, when Sriman Mahaprabhu was in His youth, Sri Advaita Acarya went to Santipura and also started explaining the meaning of visate tad anantaram as “aham brahmasmi: all souls will merge in Brahman.” Hearing of this, Sri Caitanya Mahaprabhu went to him and pulled His beard and beat Him, until Srimati Sita devi, Advaita Acarya’s wife, came and protected Him.

Prabhupada also said the same thing:

The Panca-tattva can all be golden. The Deity of Lord Caitanya is very nicely done so there is no need to change the color. Yes, Srivas Pandit has sikha. Vaisnava must have sikha. Advaita Prabhu has a full white beard. He was an old man. He was practically older than the father of Lord Caitanya. He was an elder gentleman in the town of Navadvipa, elder of the brahmana community. ALL SHOULD WEAR TULASI KUNTI BEADS, NOT LESS THAN TWO STRANDS. THREE, FOUR STRANDS OR MY GURU MAHARAJA HAD FIVE STRANDS. Only Lord Caitanya and Lord Nityananda wear crowns and nosepins. Lord Caitanya and Lord Nityananda can be decorated with all kinds of jewelry. - Letter to Govinda Dasi November 20, 1971

Advaita Acarya is not Visnu.

Sastra pramana:

Gaura Ganoddesa Dipika

by Śrila Kavi Karnapura

76.

Lord Sadāśiva, who formerly appeared as an avesa-incarnation in Vrajabhumi,

appeared in Lord Caitanya's pastimes as Lord Advaita Gosvami, who is nondifferent from Lord Caitanya Himself.

I know my reply to this is out of date, but I felt I should throw my two cents in anyway.

The verse quoted from the Caitanya Bhagavat uses the word Dadhi. This Bengali word can mean either chin, or beard. In this light one should understand that the verse can as well be interpreted to mean "placing his hand on his chin...." If we are unsure how we should interpret this word, we should look at it in context to how those in the direct line from Advaita Acarya worship him. Advaita Parivar always worships without a beard. A beautiful example of this is the lovely Sita-Natha temple in Vrindavan.