*FOUND* My brain was crosswiring two or so anime, by searching OVA releases year by year I found it, though it wasn't precognition or anything or really time travel, it was supernatural/god beings.
Title is: Guardian of Darkness, AKA Takegami War God.
This had some guy or girl who could sort of jump back like 10 min or so in time(or it was precognition), I remember something with a hill next to one of the man made rivers in Japan and something involving a kitten maybe dying? But then the jumping back in time, and one scene involving the traditional soaking baths they have.
I just remember that river and something tense happening and maybe the guy/girl falling in and I don't remember if the kitten if there was one did get brought back. The bath scene I think they got interrupted or ended up if it did have time skipping(a lot of things mix together for me) falling back in to the bath, or a precognition that they had in said bath.
I know it wasn't Devil Hunter Youko(I remember seeing it around the same time).
My mom worked at Blockbuster when I was a kid so she'd often bring home anime for me, not all of which was age appropriate.
I don't know if it came out in the 80s or 90s, but I saw i maybe around 1998 or 9 at the latest.
*ETA* He might have had something talking to him through his head? I also know it took place in basically modern to the era Japan, and him, if it was a he, was in high school ranged?