What will the closest distance be for head to toe vertical portrait using a 135mm lens?
The answer will depend upon the camera format. On 35mm/full frame it’s a telephoto lens which is very well suited to head and shoulder portraits from about 10 feet away. It’s less than ideal for a head to toe portrait though as you’d need to be about 20-25 feet away; a 50mm at about 10 feet is OK for that but lacks the flattering perspective of mild telephotos in the 90mm to 135mm range. On medium format it’s a modest telephoto lens good for those same head and shoulder portraits from about 8 feet away. With APS-C format cameras it’s a genuine telephoto lens. Not really suited to head and shoulder portraits at all. On Four Thirds/Micro Four Thirds it’s verging into super-telephoto range and completely unsuitable. So whatever the format, it’s less than ideal as a head to toe portrait lens, but may be OK if you’re using a medium-format camera at about 12 feet away. A 70-90mm lens would be better if you are using a 35mm film/full-frame digital camera.
Depends what format of camera (physical size of sensor) the 135mm lens is used on. If we are talking of "35mm full-frame equivalent" (most likely) then 135mm is rather long for head to toe photography, difficult to get back far enough from the average adult. You would do better with a 50mm, or 70mm at most, ad even 70mm would require a fair sized studio, or a good space outdoors. On a 5 x 4 inch view camera using cut-film in the old days, a 135mm lens is plenty short enough for whole body photography (180mm or 200mm is better) but I assume that you are not entering that sort of way of operating.
How tall is the subject and what camera are you using? The angle of view is 15.2 deg so on a full frame camera someone 6ft tall would need to be 22.5 ft away. 5ft tall and it’s 18.75ft. distance = (height/2)/tan(15.2/2) For a crop sensor you can multiply the distance by the crop factor (1.3 or 1.6 or whatever) to get the distance with that combination.
The best way to find out is by Trying It. You don't really need to know before buying.
The minimum focusing distance of a 135mm lens is around 1 meter. It doesn't matter whether the lens is pointed horizontally or vertically or whatever.