So, as far as time travel shows are concerned, we have at least a few choices with main characters going backwards in time, from the present to the past, but does anyone know ones with a main character that goes forward in time, either from the present to the future or the past to the present?  Preferably a large amount of time, not something like just a couple years (unless the world changed drastically in that couple years)

Off the top of my head:

The Crossing Hero (Taiwan). 
Thousand Years of Love (Korea) but it's an old drama and it really shows it's age and I don't think it's very good.

There are a few where Japanese samurais get transported to present day.

Tenchu (I think I could only find this on Dramafever but it should be free to access?) - female assassin/warrior travels to present day.

There are definitely a few others but I forget there names. Maybe I'll try looking a bit later.

Korean:
Rooftop Prince (transports 300 years into the future)
Queen Inhyun's Man (300 years too; he goes back and forth)
Tunnel (30 years into the future; he also goes back and forth)
I haven't watched it myself yet, but I think an ongoing drama  The Best Hit (highlight to see the name as it could be a potential spoiler) has a character travelling from 90s to our time.
Hit the Top. Main character jumps forward from 1993 to 2017. 
Tomorrow With You (I didn't finisht it yet) The protagonist has the ability to time travel when he takes the subway. He goes back and forth in time. (2 years into the future, I think)
In a few weeks, Live Up To Your Name will start airing. It's about a Joseon-era acupuncturist who travels forward in time and meets a surgeon. 
From the past to the present:

Tunnel
Queen In Hyun's man
Rooftop prince


I know you didn't mention it but from the future to the present:

Juunen Saki mo Kimi ni Koishite
Toki wo Kakeru Shoujo