First step is to create a custom context menu item. I do not know what type of an option you prefer but it would be the easiest to create a submenu.
Here is how the structure can look like https://jsfiddle.net/oc4e2p1c/
Then you need to call instance.getSelected()[1] which will return a column index of currently selected column. It will allow you to change styles only for the column.
EDIT: I have tested the example and it seems like the submenu is not a perfect pick, there’s one issue at https://github.com/handsontable/handsontable/issues/3550 that can cause some unexpected console logs.
However the input type color can also do they job https://jsfiddle.net/rk4eryfh/