Naruto Ultimate Ninja has two single-player modes, "Hero's History" mode and "Ultimate Contest Mode". Both could be described as story modes, with "Hero's History" mode being an extremely abbreviated adaptation of the Naruto manga's storyline and key battles, and "Ultimate Contest" mode depicting an original story. There is nothing like a traditional arcade mode, so you if you want to play the game, you only really have these two modes and the versus mode. As a whole, I found these choices to be pretty lackluster.
First off, when I say that "Hero's History" mode is abbreviated, I really mean it. This mode supposedly covers the entire first part of the Naruto manga and anime, but it cuts out a very large number of events and battles, so the entire thing can be settled in about an hour of play. Most of the characters in the game don't even show up in it at all, the focus is very clearly placed on Naruto Uzumaki's battles to the exclusion of most others, and there are no branching paths or alternate versions of events at all, so it is fairly unimaginative and pretty much has no replay value. It is obvious that this was not the main focus of the game.
"Ultimate Contest", on the other hand, is fairly elaborate. Rather than being a linear series of events, it lets you freely control Naruto as he explores the Hidden leaf Village and participates in the original story. This could have been really fun, if it were not for the fact that the original story is a fairly simplistic tournament plot and you really have nowhere to explore except the village. The plot really is just a shallow excuse for a number of battles against other ninja, which would be fine (this is a fighting game after all), if it were not for the fact that the set-up limited the player to using Naruto almost exclusively, made him fight pretty much nothing but the other Leaf and sand ninja that are his own age, and placed all of the battles in an extremely limited sub-set of the game's many fun battlefields. So, the main plot of the "Ultimate Contest" mode has no real variety and a boring plot.
In addition to the tournament plot, "Ultimate Contest" mode features a number of events and missions you can undertake, which help you gather money you need to unlock bonus material (voice samples, movie clips, etc) as well as expand your repertoire of jutsu for the customization system. Fortunately, you can actually use any ninja you have unlocked for this mode in the event and mission battles, but unfortunately the events and missions don't really focus on battle at all. An event is just as likely to be a mix of completely brainless fetch quests (AKA scavenger hunts where the items you need to find show up on your map) and minigames as it is likely to contain a battle (and battles are often combined with fetch quests and minigames). What is more, most of the rarest and most important items you can acquire, the ones that allow customization of each character's strongest Ultimate Jutsu, never require battle at all. Some of the challenge battles found in this mode can be fairly fun, but as a whole, despite this being a fighting game, there simply isn't a lot of fighting, certainly nowhere enough to let the player try out all of the game's characters, Ultimate Jutsu, and customization options.
"Ultimate Contest" mode also has an experience system that lets ninja level up through battle and completing missions, but this system never really seems to matter (most battles can be won as easily with a low-level ninja as a high-level one). Actually, since other than the battles where you have to use Naruto there are practically as many battles as there are characters and most of the time every character gains experience at the same rate, there really isn't a lot of purpose to trying to raise levels for anyone.
The one thing I want out of the single-player modes of any fighting game is a chance to fight a number of battles using all of the different characters in the game. In both original fighting games like Street Fighter, Soul Calibur, or Guilty Gear and high-quality licensed fighting games like the DBZ Budokai series, a variety of arcade modes and versatile story modes fill this niche very well. Despite the strength of its battle system, Naruto Ultimate Ninja 3 just doesn't have this as an option, and thus falls flat. At least it has a decent versus mode...