What is Mugen chewing? What is Mugen chewing always sometimes he chews small sticks, some other times he chews long stems with leaves. Is that Japanese style? and what is that plant name?
Anime characters chewing on a blade of grass/toothpick is considered a trope in anime and is usually attributed to the delinquent types.


  Sometimes a character will suck or chew on something as a form of
  characterization. These range from toothpicks and blades of straw or
  grass to lollipops and cigarettes.
  
  In anime, having a piece of straw in one's mouth is a common depiction
  of a banchou, or juvenile delinquent gang-leader character.


Some other characters that have been noted to conform to this trope:


Tetsuya Kusakabe from  Katekyo Hitman Reborn!, a delinquent character.





Ikki Takeda from History's Strongest Disciple Kenichi, who was formerly a member of a gang called Ragnarök.





Mifune from Soul Eater, who's personality is described like this:

Often stoic in nature and staunch in attitude, Mifune tends be somewhat quiet in situations that do not directly involve him and remains mostly serious as well as being straight-forward..





Ash's Treecko from  Pokémon, described as cool, calm, collected and serious.






Wikia, on Mugen:
Rude, lewd, vulgar, conceited, temperamental, and unhinged - Mugen is something of an antihero.

Thus, it can be inferred that the "blade of grass in mouth" trope is a type of characterization that mostly represents a serious/delinquent sort of character. 



The plant isn't usually of significance and isn't the same in every case unless explicitly stated. In a Japanese page that describes Ikki Takeda from the second image above, the following is written with respect to the blade of grass. 


  常に葉っぱを口でくわえて...


which translates to "Always with a leaf in his mouth...", indicating that he's not chewing on any special plant.

As for what Mugen is chewing in the particular image posted, I'd guess it's bamboo.