What words could I use to describe the event of having successfully and completely "read" or understood someone's personality, upon first meeting that someone?
Woman meets man. He asks her something out of the ordinary, but she'd been expecting exactly that sort of question, from him, despite having just met him. She smiles, because she's got him all figured out.
figure someone or something out
to begin to comprehend someone or something; to come to understand someone or something better. I just can't figure you out. I can't figure out quiet people readily.
To see sb's true colours, may suggest the idea. It may also refer to a first meeting and it is specifically about understanding the true personality of someone:
to see someone's real character for the first time, especially when it is unpleasant:
- It was only when they started to work together that she began to see his true colours.
To "get" something is to understand something; to say to someone "I get you" or "I know you" is to say "I understand you on a fundamental level" (usually intuitively rather than analytically).
For instance, in the Tom & Angela song "You Get Me", Tom is not saying "I understood your question, Angela", but something more like a sense of "immediate, complete, and intuitive" understanding.
To extend the metaphor to "within a short timeframe", we'll usually add a temporal qualifier, such as "I got you instantly" or "I got you the moment we met". You could also say "I really got his number" (or, more rarely, "his measure"), though the connotation there is more adversarial ("he's clever but he can't outwit me", or "I know how we can pull one over on him").
Some alternatives include "I read you like a book", and "saw through you", but again these have slightly negative (or at least self-promoting) connotations.
As a last resort, if your new friend is a sci-fi enthusiast, you could always say "I grokked you".