Shinjuku to Tochomae

Hihi,

I'm arriving at 1pm, and leaving Tokyo at 11pm by Willer Bus (at Sumitomo WTB building). I'm looking for things to do in Shinjuku for half a day, and happen to see a map for a half day trip on japan-guide. The map shows a red portion leading from Shinjuku station to Tochomae.

I was considering walking there, but is the red area an underpass? Can i access it and walk all the way via underpass to Tochomae? I'm dragging a luggage around so I thought perhaps an underground route at that hour in Shinjuku may be more forgiving than walking around in the open air (plus its february when im going, so it be quite cold?)

Would this be advisable? Willer Bus gave me a good guide to walking there via main roads, but i was just thinking if this would work too, as im planning on having dinner around Shinjuku, maybe walking around Takashimaya or something around the station before visiting the Tocho Building before boarding the bus?

here you go
https://willerexpress.com/x/bus/dynamic/3/en/html/map/tokyo/willer_ter...