>一番記憶に新しいのは
I've always understood 「記憶に新しい」 as an idiomatic set phrase so I've never given much thought to its grammatical constitution.
If pressed, I'd say that the に of the 記憶に新しい has a similar meaning to the "in" in "fresh in my memory". A similar usage is 記憶にない (it is not in my memory = I forgot).
>過去を旅することができることは、それだけでもドキドキすることがたくさんある
それだけでも here means "by itself", or "in and of itself". As already pointed out, the intended meaning in this context, roughly speaking, is that " time traveling is exciting in and of itself, even if anything special doesn't happen"
>>The ability to time travel has (involves) many things that, in and of itself, makes one's heart beat (gets one excited)"
>僕も同じので(お願いします)。
I am ignorant about the precise grammatical details, but it might help to think of it as a short-hand form of 僕も同じものでお願いします.