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.

One excellent movie I forgot to mention is Ponyo (on the Cliff by the Sea) 2008.

A delightful fantasy about a magical fish who turns into a girl to be with the boy she loves.

Ponyo is reminiscent of the old Disney animated movies of the 50's and 60's in style, storyline and artwork. A lovely adventure!