My daughter just turned a month yesterday, is it ok to go outside w/out a cover over her face?

of course. fresh air will be good for her. Just keep the sun off of her skin :)

if it's sunny I would just put her on a hat if you have one or a light blanket. Sunblock isn't reccommended for babies under six months old.

If it's sunny you should probably put a little baby sunblock on her since babies can burn super easy