We often like to immediately leave a country capital for the countryside or smaller cities, even on arrival day. But, it can always be unpredictable if your international flight arrival is delayed, navigating metro out of airport is slow, or you get briefly lost in the main airport or train station.
So, is it risky or unrealistic to wait to buy AVE tickets (or other high speed train) only AFTER arriving at Atocha train station? I don't know if seats truly sell out often in very late May or early June, or how much more expensive a last-minute purchase is vs. buying weeks in advance? Trains do depart hourly for this route. If you DO miss your reserved train, can you apply a refund towards a later train that day?
If this is completely unwise, do you have tips for a jetlagged half day near Atocha station? Parks, a great cafe/restaurant, or short walking tour are more realistic with jet lag vs. visiting a major museum or site, probably, being so groggy. We speak Spanish if some kind of tour/site is quite short.