The Arab Spring protests began in December of 2010, and their first main impact - the removal of Tunisian leader Zine Ben Ali in a military coup - occurred in January 2011. On February 11, 2011, Hosni Mubarak resigned, and in late February the Libyan Civil war began. The Syrian protests and subsequent Civil War began around March-April, and the Libyan Civil War concluded in October.
This means that, if these events were shifted exactly 3 years to the past, they would have played a huge role in not only the last year of the Bush government, but the 2008 Presidential Election as well. Let's not forget that the US still has troops in Iraq in 08.
How would Bush respond? My guess is that he would act very similarly to Obama. In Tunisia and Egypt, the US would tacitly support our allies before switching sides and endorsing the activists once it was clear that Ben Ali and Mubarak were on their way out. In Libya, we would wait quite a while before intervening. People in 08 would be wary of another foreign war under the leadership of Bush, but intervention in Libya was ultimately a guarantee due to the significant desire to intervene by Britain, France, and other European nations.
Let's also remember that 2008 saw the beginning of the Recession. The financial world was beginning to panic as early as 2007, but September 08 saw it really fall apart. This means that in early 08, economic issues won't be as big of an issue, and foreign policy could very well define the primaries and the general election until the big banks really start falling in September.
Ultimatly, I am pretty sure that Bush would act very similar to Obama. What I am not certain of is how the presidential candidates would act. Perhaps Hillary could use her foreign policy chops to try to upstage Obama? I think McCain will still secure the GOP ticket, but how would he face an Obama or Hillary? I really don't know.