I often see on the Internet one sentence, He was rich (He is not rich at present). Why does the sentence have such an implication?
Because was is the past tense of the verb to be.
The verb to be is used to link the subject’s state with an adjective (or some other part of speech.)
The tense of the verb contains the information to tell you when this state existed:
I am ill = I am currently in a state of illness
I was ill = at some time in the past, I was in a state of illness.
Is it customary only for this sentence to express that meaning?
Yes. The tense of the verb is usually enough to indicate the time that the state occurred.
What if I want to say he was rich in the past, but also rich at the present? Should I add some time point to make the tense clear" like he was rich when he was young but still rich when grown up.
You can use the present perfect with an adverbial phrase of duration: “He has always been rich.
See another example, I didn't sleep well yesterday.
To sleep is not a verb that gives the state of its subject: it is a dynamic verb. The same guidance applies - the tense tells you when the action of the verb occurred:
I sleep = I am currently asleep
I slept = at some time in the past, I was asleep.
I didn't sleep well yesterday? (Who knows if I can sleep well tonight?)
Nobody knows... and neither do you. But you can use the present continuous form:
"I'm not sleeping well" this indicates a recent and unfinished experience and implies that you expect "not sleeping well" to continue.
If you have not been sleeping well for years, then you can use the simple past:
"I do not sleep well."
For very useful information on tenses, see EL&U’s question: How do the tenses and aspects in English correspond temporally to one another?