Is There a Way to Play the Original Donkey Kong on the PS4?

Donkey Kong was originally created by Shigeru Miyamoto, and the rights to the video game still belong mostly¹ to Nintendo.

As such, no original version has been made without direct authorisation from Nintendo, including on competing console or web. The version you played on the internet was likely a clone, or a cheap copy. You can not own the rights to a game idea; If the game is similar on mechanics, alone, it is not enough to declare it a copy of the intellectual property.

This is a common practice for Nintendo, as games can be released "exclusively", meaning that if you want to play the new games, you will have to buy their new console.

You might be able to work something out through emulation, however, this takes you into a questionable grey area in terms of legality. It is common place for each game to include a "Terms of Use" that strictly prohibit said actions, but this can often be overridden by your residing countries legal definition of 'Fair Use'.

_{¹ In terms of the original game, it could be argued that Nintendo, themselves, do not have complete rights, as detailed in a Gamasutra article on The Secret History of Donkey Kong. They outsourced the original code to a separate company, who later successfully sued them for its reuse in the sequel, Donkey Kong Jr.}