The shuttle bus from Southampton Parkway Station is timed to depart just after main line trains and will usually wait if they are running a few minutes late but that isn't guaranteed. The time from Southampton Airport to Whiteley can vary widely at that time of the morning so you probably need to confirm with HR that they are happy for you to travel on that bus and will make allowances if you it is delayed if you are going to take that route.
As Peltonlevel and TMT say it is a brisk 25-30 minute walk using the traffic free footpath route from Swanwick station and you need good directions to go that way. You can knock 5 minutes off by going the suicide route across the M27 roundabout at Junction 9 but at that time of the morning it may take you some time to cross the first slip road which has a continuous flow of fast moving traffic with no traffic lights to create a gap.
Taxi from Swanwick is the best bet as long as there is a taxi at the station. Peltonlevel can probably advise as to how likely that is.