Every job can solo fairly well, but much of it is dependent on getting the better cross-class abilities for survival. Level Pugilist to at least 6 to get Second Wind
also leveling Conjurer up enough to get Stoneskin, Cure, and Shock Spikes will go a long way in helping any class survive. Also, once you decide on what job you like, consider a crafting class to support that. Be it a blacksmith for making and repairing your weapons or a clothcraft/leatherworker/armorer to keep your body armor in tip-top condition.
While you are interested in solo play, many people are willing to join up on a couple of leves if you find yourself with only 30min-1hr to play around and just want to make a little progress. Also make sure you join up with random people for Behests which happen hourly at all the Aetheryte camps. It's great SP and you might make yourself some friends in the process or find a person who wants to stay to do a couple of guildleves.
As for best solo class, I really couldn't answer, I have a fairly easy time soloing on Pugilist but I also have a decent amount of cross class abilities and good gear so your mileage may vary. Don't get trapped into thinking you need to pick a class based on perceived solo abilties, with a small effort you can up your survivability on almost any class