By old style, I'm guessing no fancy CGI or high tech looking artwork.
Probably like Fire Tripper 1985 or Princess Mononoke 1997
I hope you don't mean stuff like Flying Phantom Ship 1969, or Alakazam the Great 1960 ;)
Some that may fit the niche are:
A Letter to Momo 2011 - Slice of life, spirits
Howl's Moving Castle 2004 - Fantasy
Koe no Katachi - A Silent Voice 2016 - Slice of life
Patema Inverted - Sakasama no Patema 2014 - Sci-Fi, adventure
Spirited Away 2001 - Fantasy, spirits
Summer Wars 2009 - Slice of life
Sword of the Stranger 2007 - Edo-era style, action, adventure
All of these are fine movies that rely more on storyline rather than special effects. Well worth watching.