You shouldn't be having problems soloing with Nasus. If my team has a jungler I always ask for the side 1v2 lane and typically wind up controlling it unless I'm facing a particularly nasty combo.
If you are having trouble with survivability/harassment, grab Dorans shield as a starting item. Yes rushing sheen is fun but it's not the most practical thing for the solo laning phaze. Especially if you are grabbing teleport... think about it, you can go back to buy and TP back to your lane at any point... so are you more concerned with having the extra mana from the crystal to spam or more concerned about not giving up early deaths? Remember you are just going to blue pill back and buy when you are oom or have enough gold for the an item trip, and then TP back without missing a creep wave. So grab the dorans, it's great on Nasus.
To further help with survivability, once you are high enough level/have the IP invest in some flat HP quints. I also advocate the clarity glyphs for mana regen. To finish out your rune page go with armor pen reds and dodge or hp/lvl yellows.
While Nasus is a great laner... its not because he is really tough or a good early harasser or can put out a lot of damage. It's because of his zone control and farming ability with spirit fire. It sounds to me like you might be a little too worried about engaging the enemy in these early levels. Ignore them. Learn to avoid their harassing techniques by staying out of range when you don't need to be in it, behind creeps to avoid skill shots, etc. Get your 1 point in SS and last hit with it whenever possible. Remember that soloing is a balancing act of keeping the waves equally pushed so that you can maximize last hits, unless you have developed your own personal strat for effective last hitting of creeps under your tower (1 hit, 1 tower hit, SS last hit on melee, etc). Drop SF on the enemy if they are standing with their creeps, or whack with SS when they come up to last hit, but don't worry so much about trying to kill them or send them to the well unless you really think you have the opportunity to pull it off. You don't need to SF every wave and push your enemy to their tower, then they are just soaking up free XP safe from you and possibility employing the same tower stealing last hits you are.
Nasus has such solid solo mid capabilities that you can easily get 2 SS hits per wave (3 or 4 kills total), as well as both wraith patches. To get the wraiths just SF the creep wave and push it hard so you have a few creeps advancing on the enemy tower, run to the wraith patch, drop your lvl 3 SF, wait 2 tics, last hit the big wraith with siphon, and run back to mid in time for the next wave. If you are controlling both wraith packs and not missing any creep waves you will be out-leveling your opponant.
Also abuse your ult and take down dragon quickly. If you don't sense any team battles coming up early, you can solo the dragon with your ult, dorans, and sheen. It's best to grab golem buff first because it will reduce the CD on all your spells (including bringing your ult back online faster). Grab golem, then rape the dragon (use all ur spells with the sheen proc), blue pill to the well, and tp back to your lane, should only have missed a couple waves. Can do the same with golem.
All this being said, I'm not a strong advocate of Nasus soloing mid. Yes, you can easily handle it. No, you are not the best choice. You should have a better carry hero taking mid. Nasus can be a powerhouse but at higher ELO he is played as a tank. Dorans, Merc Treads, Aegis, TriForce, Banshees. Nasus is much more effective when he outlevels the enemies, but it's more important for your carry that will be raping the team to be farmed and leveled and ganking. For nub pub stomping, go nuts. Just don't expect to be soloing Nasus mid much in higher ELO.
Hope you can take something useful out of this. GL