If your actual end point is well to the west of Southampton (e.g. Exeter), rather than being Southampton itself, you could fly from Newcastle to Newquay with Loganair. (It's a new service, starting on 1 April).
Coach travel, with National Express, takes between 11 and 12 hours from Newcastle to Southampton, with a change at either Birmingham or London. The fares are cheap though, at under, or just a bit over, £20 one-way.
Megabus has a
direct coach service though, taking 10h 15m, with a typical one-way fare around £17.
The National Rail website seems to be throwing a wobbly at the moment but, by trying some sample dates and times on the LNER website, I'm finding the one-way outbound fare as low as £52 (which involves getting across London) or £88.10 (direct), with journey times in the region of 5½ hours. Coming back, I'm again seeing £52 via London but just £70.70 direct.
https://www.lner.co.uk