The Girl Who Leapt Through Time/WMG

The opening scene wasn't just a dream; Makoto was either killed or severely injured by the baseball hitting her in the head.
This made to go back in time to save her, which in turn is why. Later in the movie

Makoto unknowingly ends up paying back during the course of the movie, when.

Makoto will be the inventor of the time travel device
Early in the movie, she's considering whether to study science or not; her aunt says that she's not the type to just wait for a guy, she'll go find him; she tells Chiaki at the end that she'll "come running" to meet him in the future, and finally, at the very end, she tells Kousuke that she's made a decision. The decision? To study science and eventually invent a little walnut that charges you up to travel in time!

You could even take it a step further and say that Chiaki's real reason for traveling back in time was to be the impetus for Makoto to invent the device, thus creating a Stable Time Loop.

I interpreted it rather differently. Her aunt, or rather "aunt" is person from future sent by Makoto/her coworker/her descendant (I doubt she is future Makoto) to preserve this painting. She promised that she will take care of it, after all. It explain also strange reaction to her story, telling about leaping, encouraging Makoto to leap etc. This was of course to start this whole chain of events that will end in sending "aunt" to future. And decision, by the way, was to be historian and preservator.
 * Jossed, Makoto's aunt acquired the ability to time leap in more or less the same way she did. Furthermore, she met the inventor of the Time Travel Drug, he wasn't Makoto.

Chiaki's Time is Less Than a Decade in the Future
Thus having Makoto wait for him be not quite such an incredible task. Baseball only disappeared because the Boston Red Sox won another World Series, which coincidentally caused a low level apocalypse. Time travel was invented as well.
 * If baseball disappears, it will not be because of a third 21st century World Series title for the Red Sox; it will be because the Red Sox lost the World Series to the Chicago Cubs.

The painting survives to Chiaki's time as a result of him traveling to the past.
In the original unaltered timeline, Makoto dies by being run over by the train, and the painting vanishes at some time thereafter. Chiaki's trip to the past results in Makoto avoiding her fatal train accident (as a result of acquiring Chiaki's time travel ability), and her relationship with Chiaki spurs her to succeed her aunt as caretaker of the museum's art. Makoto's work then allows the painting to survive to Chiaki's time.

Aunt Witch is Makoto
The timeline begins. Makoto befriends Chiaki. Their feelings for one another grow. Makoto finds the time travel device. She has fun messing with time for a while, until she realizes she's out of leaps. Tragedy strikes. She has to watch her friends die because she couldn't save that one last leap to keep them alive. Then everything stops. Chiaki approaches her and tells her everything, including that he used his last leap to save Kousuke and his girlfriend. Unfortunately, Now that he's broken tardis protocol, he must dissipate into the void, or go back to the future to face trial for temporal crimes or something, and what's worse, he never got to see his painting. It's okay though, because Makoto has one more leap, right? Wrong. She doesn't have one more leap, and the love of her life is gone forever. She never planned to wait for him. Jump 15 years. Time travel is invented. She finally has the chance to save Chiaki, but she's too old to do so as herself. So she goes back, further than Chiaki. She goes back to around the time of her birth. She befriends her own mother and receives the affectionate nickname "aunt witch". Now she has an in. She can help herself by convincing young Makoto that her frivolous leaping may be hurting someone else, perhaps just enough to give her that one last leap that she need to save Chiaki, (if a boy is late in meeting up with you, you're the type to go look for him). But why? Why did she go back so far? Why did she have to be there so long before she even met Chiaki? Simple: she wanted to give him his painting. She needed time to hone her craft, make a name for herself, and use that name to get a job as the restoration artist in a high class museum so that she could restore the painting that Chiaki traveled so far to see.
 * Jossed, Aunt Witch has been outright stated to be Kazuko Yoshiyama, the protagonist of the original novel the film was 'based' on (though it was more of a spin-off sequel of sorts). She experienced a similar course of events to what happened to Makoto in the film though, which is why she knew about time-leaps.

Makoto isn't travelling through time, she is hallucinating/her memory/perception of linear time is broken
Did you see just how many times during the movie she bangs her head against hard objects? Something in her brain was bound to get damaged at some point.

The Japanese language evolved greatly by Chikai's era
IIRC, they mention at some point that he cannot read Kanji at all. One explanation could be that in the future he came from, Kanji isn't used anymore. Languages change all the time, and we don't know how far his future is.
 * On the other hand, if people stop being taught Required Kanji before their last years of high school once computers become the de facto method of teaching and doing homework, it's possible that Chiaki is from as near a future as the early 2020s, and the Japanese government has started full-immersion teaching of using a system of hiragana/katakana with an identifier code for homonyms. Since Japan mostly uses kanji based on the Chinese pronunciation of the symbol rather than the Chinese meaning of the symbol and Japanese pronunciation, it seems perfectly reasonable that the government (in the future of 2006, or of the year of whichever adaptation first introduced that detail) would decide that Japan would trail behind purely letter-based countries in computer efficiency and therefore ability. Said government then sets up a system wherein Japanese students learn the new system, which consists of all the regular hiragana and/or katakana, but instead of kanji mostly uses the words spelled out in kana with a code to identify which homonym is being used beyond simple context.

Chiaki lied about why he .
The painting was probably important, but, whatever age she was, told Chiaki to lie so that through him, she could create a Tricked Out Time loop.