Original Celly's run hotter than CII's. This might make you think, "Kewl. I'll put the old HSF and fresh thermal paste on that lil' glass slug, and Hi-Oh Silver."
The Alpha 6035 will make the transition safely. You're OK.
My Vantec SI5, even with AS and the new Delta 38 on it didn't. After toasting the PIIIE 550 with five brief excursions to 56C, it took it five months of 500 spontaneous reboots, and finally a dead CPU before I picked up the clue phone.
Shuh! It's the HS, stupid. Even though the FC-PGA's put out less heat than the Celerons, the heat they do produce must be sucked away from the little glass slug as fast as is possible! A HS with some heft to it is necessary. The SI5 is a weenie little thing, even with the monstro fan, it was not up to the task. You may want to get a Delta 38, for the summer. A 60 mm to 80 mm fan adapter will let you miss the "glory" of the sound of the Delta 38.