Hi Joyce - The ferry you are going on is often advertised in UK as being the "Newcastle-Amsterdam Ferry" but of course it neither goes to Newcastle nor to Amsterdam - and North Shields is hardly closer to Newcastle than IJmuiden is to Amsterdam!!!!
9 til 5 is tight for what you want to do. The Newcastle metro is an excellent facility EXCEPT that the DFDS terminal is a 1.1 mile walk from the nearest station at Percy Main.
So you have a choice – use the dedicated DFDS busses directly between the terminal and Central Newcastle (at £3.50 each way) or use the Metro with its walk to the nearest station
Making sure you get back on time is obviously important. I see that the last bus direct to the ferry terminal leaves Central Newcastle at 15.45. I would have thought that was the "best bet" to make sure you get BACK to the ferry in time – albeit perhaps a bit early. You always have an (expensive!) taxi as a Plan B if you miss the bus or a mixture of metro/taxi!
I think you need coins to buy a metro ticket via machines at the unmanned stations -see
http://www.nexus.org.uk/metro/ticketsWallsend is close by North Shields and does have some remains of Hadrians Wall. What you would see is a mere shadow of what you would see along the main part of the wall further west of course – but if you just want to "tick off" the site then you could, immediately on arrival, walk to Percy Main station (this would save you waiting for the dedicated bus to leave which could well be half an hour after the boat arrives anyway –which would just be "wasted time") and take the metro towards Newcastle for 3 stops getting off at Wallsend. Only 200 yards away south is the Roman fort of Segedunum. It costs £4.35 to visit the Fort, Baths and Museum (which seems pricy to me for what there is but then I hate paying for entrances)! It has a web site and an entry on Wiki. Say you arrive by opening time at 10 am. You could hardly give it much less than 1 hr? And then you have to get back to the metro and on into the city. With luck you could be there by 11.15?? Which leaves you 4.5 hours to get to/from Durham and see that city.
Trains to/from Durham from Newcastle Central run every 15 minutes or so and take around 15 minutes each way - so, at worst, that could take around 45 minutes return. Get an OFF PEAK DAY RETURN ticket for £5.80. On arrival check the last train you want to take back from Durham to get you in to Newcastle in time for your bus and pace your visit from there (with the experience of walking to Percy Main station you could even consider taking the metro back to it). The almost 1 mile walk from Durham station to the cathedral will take around 20 minutes (another 40 minutes return) leaving c 3 hours for Durham itself after travel/walking time! I am not sure I would even bother to go into the castle. Spend all your prime time at the Cathedral and see how it goes from there. Some of the views of the Cathedral from the river/the bridges are very fine and, unless the weather is terrible I wouldn't miss those – they are "classic". So, 2-3 hours for the Cathedral and the walks – it all depends on how small you grind when you visit. The small city itself is nothing special. My wife was at University at Durham and agrees with my assessment of the castle by the way. I would spend spare time in Newcastle itself rather than the castle! I think that Grey St has more listed buildings than any other street in England. See
http://en.wikipedia.org/wiki/Grainger_TownThen there are the Quays at the bottom of Grey St, walking under the "original design" for the Sydney Harbour bridge, the modern bridges/the Sage arts centre etc etc.
If you are REALLY only wanting a "tick" for Hadrian's Wall for any old bit of Roman stone then you could miss out on Wallsend/Segedunum and make for Benwell which much closer to the city centre and free to visit (
http://www.english-heritage.org.uk/daysout/properties/benwell-roman-temple-hadrians-w all/ ). It has a small temple which is right in the middle of a housing estate 1.8 miles west of the Central Station along West road – a couple of bus stops.