Not exactly.
The draft war room shows the time your pick will be selected.
The player at the top of your draft board (on the right hand side) will be the player you select.
You can manually edit this by adding players to the manual priority list (middle section). The players on that list will automatically be at the top of the draft board.
Also, you probably want to Edit the draft board settings to make sure a player you would want is selected in the event for some reason your pick comes and you don't have anyone on the manual priority list.
TIP: set P and K weight to 0 in order to make sure you don't take one by mistake.