Wasn't Anakin's "on time" arrival just a mere coincidence?
How do you know that Palpatine had expected Anakin's arrival?
Well, its not the most canon or canon, but in the various books i've read its made very clear that Palpatine pretty much orchestrated everything. From Gunray taking over the Trade Federation and then attacking Naboo. To Sifo Dias ordering the clones etc. So that he orchestrated an attack on Coruscant to get Anakin back to the Capital and on Kashyyyk to draw Yoda away(since he had the best relations with the Wookiees), makes sense. That leave Mace as the Senior Jedi council member on Coruscant, the Jedi Council member who had the gravest doubts about Anakin. Then there's the whole business with Palpatine appointing Anakin to the Council, inflating his ego, knowing that the council would at best grudgingly accept Anakin, causing insult to Anakin. Then Palpatine could reveal himself to Anakin, knowing he'd turn Palpatine in, but also knowing that the Jedi Master who doesn't trust Anakin, Mace Windu, wouldn't bring him along on the arrest. This would further insult Anakin, who would rashly act, and show up "just in time".
Palpatine wasn't being defeated, as evidenced by his miraculous recovery once Mace was down to one hand. So he was clearly pretending to be beaten, and the reason he would pretend to be beaten was to draw Mace in, knowing that Anakin was riding (unknowingly) to Palpatine's rescue.
I think there is also the implication that Palpatine had some ability at foresight. I can't recall the exact lines but in both ESB and ROTJ doesn't Palpatine have lines about how things are "proceeding according to my design" and that he has forseen that Luke would be a powerful ally. Vader even mentions that the Emperor has forseen that he and Luke could overthrow him. So I think Palpatine has forseen that Anakin would be forced to decide between saving him, or allowing Mace to kill him. Therefore he just has to fill Anakin with lies and doubts to make sure he decides on the result Palpatine wants....
