The Arab Spring in many ways was driven by the cascading population pressures in the Arab World (Egypt in particular) and was ultimately triggered by food insecurity. If you have food insecurity be a big deal in 2001, its possible.
The big issue of course is that the Second Intifada was occurring at the time, and would inevitably be dragged into the Arab Spring rhetorically or otherwise.
Keep in mind that all of this was before George W. Bush's Freedom Diplomacy, which began in his second term, took hold, and ideas like the Town Square test and tying military aid to democratic progress had not yet begun. This would therefore embolden the autocratic leaders from making any damaging concessions prior to the Arab Spring, and would allow them to take a harder line on the protests.
Bashar al-Assad would be very new to power and perhaps would be ousted in a coup, as he would not have developed the kinds of relationships he did over the 10 years before the Arab Spring OTL and would have looked quite weak.
Mubarak probably would clamped down on the protests with more force and perhaps could have stayed in power.
Saddam Hussein, who had wrapped his regime in the colors of Sunni Fundamentalism in the late 90s (remember the Blood Quran? The purge of secular Republican Guard officers?), would have quickly turned any protests into an ethnic competition, and would have slaughtered the Shiites again, while the Kurds would have been protected by NATO airpower.
The French would have ensured nothing too drastic happens in Tunisia. The Algerian Civil War would not be impacted.
Lebanon might have an expedited Cedar Revolution. If Hezbollah seizes power in the ensuing vacuum, it is possible that Israel invades Southern Lebanon, and likely does the job more competently than OTL because of Lebanon only having 1 year to consolidate their positions, and the Sharon government being way more competent militarily than the later Olmert government.
Qaddafi would have stayed in power without NATO interference. Bush's pre-9/11 foreign policy likely would not induce him to want to intervene, and after 9/11, too much attention would be focused on Bin Laden.
The role of AlQaeda is interesting in this scenario. They were not the pariahs that they were after 9/11 as of yet. Its possible that they could have been big influencers on rebel movements.