If someone did papa karma and got punishment for his karma on the earth. May be jailed for entire life, or death punishment or any type of punishment, then will he be again punished for that?
No, A person can't get punishment of same sin twice. After getting punishment of his sin, person becomes sinless. If a King punish a culprit for his sin Or he shows mercy on culprit, in either case that person will be free from his sin. But if King don't gives suitable punishment to that culprit then King himself has to suffer from that punishment (which was needed to give to the culprit). Following Ramayana verse saying the same thing. When Rama was teaching Vali about Dharma...
राजभिः धृत दण्डाः च कृत्वा पापानि मानवाः |
निर्मलाः स्वर्गम् आयान्ति सन्तः सुकृतिनो यथा || ४-१८-३१
maanavaaH = humans; paapaani kR^itvaa = sins, on making; raajabhiH dhR^ita daNDaaH ca = by kings, imposed, punishment, also; nirmalaaH sukR^itinaH santaH yathaa = unblemished - sinless, with good deeds, pious souled ones, as with; svargam aayaanti = to heaven, they come.
When kings impose proper punishment on the humans who have sinned, they become sinless and enter heaven as with the pious souls with good deeds.' So says one verse of Manu.
शसनात् वा अपि मोक्षात् वा स्तेनः पापात् प्रमुच्यते |
राजा तु अशासन् पापस्य तद् आप्नोति किल्बिषम् || ४-१८-३२
stenaH = a thief; shasanaat vaa api mokshaat vaa = by punishment, either, even, by releasing - by clemency, or; paapaat pramucyate = from sin, will be released; a shaasan raajaa tu = un, punishing, king, but; paapasya tat kilbiSam aapnoti = sin's, that, blot, he derives.
Either by punishment or by clemency a thief will be absolved from sin, but the king who does not impose proper punishment will derive the blot of that sin.' So says the other verse of Manu.
As said in Rishabh's answer, once the King (or Government in today's time) punishes an evil-doer then he (the evil-doer) is free from the guilt.
Manu Smriti 8.318. But men who have committed crimes and have been punished by the king, go to heaven, being pure like those who performed meritorious deeds.
If the King pardons the evil-doer, even then, it is considered that the evil-doer is freed from the guilt as said in the following verse. But in that case, it is the King who takes upon himself the guilt of the evil-doer.
Manu Smriti 8.316. Whether he be punished or pardoned, the thief is freed from the (guilt of) theft; but the king, if he punishes not, takes upon himself the guilt of the thief.