Я сразу понял её намёк и перестал оправдываться.

I immediately understood her hint and stopped making excuses.

Breakdown of Я сразу понял её намёк и перестал оправдываться.

• я: I
• и: and
• её: her
• понять: to understand

Questions & Answers about Я сразу понял её намёк и перестал оправдываться.

Why is it понял and not поняла or понимал?

Понял is past tense, masculine singular, so it implies the speaker is male. A female speaker would say поняла.

It is also the perfective past of понять, so it means understood / came to understand as a completed moment.
Понимал would be imperfective and would suggest an ongoing or repeated state, not that sudden moment of understanding.

What does сразу mean here, and can it go somewhere else in the sentence?

Сразу means immediately, right away, or at once.

Yes, it can move. Russian word order is flexible. For example:

• Я сразу понял её намёк.
• Я понял её намёк сразу.

Both are natural. Putting сразу earlier gives a slightly stronger feeling of immediacy.

What case is намёк, and why isn’t it намёка?

Намёк is the direct object of понял, so it is in the accusative case.

But намёк is a masculine inanimate noun, and in Russian, masculine inanimate nouns often have the same form in the nominative and accusative singular:

• nominative: намёк
• accusative: намёк

So the form does not change here.
Намёка is a different case form, usually genitive.