Is it possible to visit Ali mountain and be in Taipei on same day i.e no staying there.

Yes, it is possible to gave a round trip from/to Taipei by visiting the mountain. I suggest you use the way of high speed train (transfer at Jiayi) + bus.

Generally, it is plesant to visit in July. The only problem is that it will be during the typhoon season, so you should pay attention to the weather forecast in advance!