I'm just throwing this out there because it's how I solo things, and I literally never see any other PUPs doing it my way. I'm not saying my way is the best way, and it certainly is not the only way, but it does work. Compared to what I see most PUPs doing, I have an extremely defensive style of soloing. But the defense pays off; I power through mobs and links that destroy most other jobs. Here are a few things I have done solo:
Soloed Tier 1 VNM from 80%-death after the sam/dnc died
Soloed several NMs
Handled a seven EP mob link (until Guiver popped on top of me... that was brutal)
There are a few simple keys to my style of soloing.
1) Full evasion gear
2) Utsusemi (except for campaign or monk nm's)
3) whm or curebot puppet, as they do not require /war to control hate
4) Avoid deactivating whenever possible
This is what I solo in most of the time. It's really not that expensive. Using the 2 agi -> 1 eva, 1 eva skill -> 0.9 eva, and the fact I have evasion bonus 3, I can say I have about +90 evasion. It is all about evasion.
NQ Oil (mainly for removing status debuffs with AF boots on macro)
Battle Gloves (yes, level 14 gear)
Feral Ring (because a second Wivre Ring refuses to appear on the AH)
Combat Caster's Slacks
I don't have Stringing Pummel, so I have two weapon skills I use. If I'm feeling lazy it's just Howling Fist. You can macro in str and vit gear, but it won't make much difference. If I'm feeling fancy I'll track my automaton's tp and when it uses Magic Mortar it's a go for Dragon Kick to make light. It's fun, but not that big a deal.
If I know I'm facing something hardcore I will pay for a jack-o'-lantern, but 99% of the time it's crab sushi.
If you are in campaign always go PUP/WAR
If you are soloing random things and not sure what to do always go PUP/NIN
If you know you will be fighting a nm that tears through shadows, go PUP/DNC
Basically, the only time I solo PUP/WAR is when I'm taking down campaign mobs and I want to get in the extra damage for more exp/hr. In my opinion it's not really a great solo job when it comes to survivability. The whm puppet, which you will use 99% of the time when soloing, does not need provoke to keep hate off of it.
The evasion gear + shadows strategy is extremely powerful, add in a whm puppet are you are a rockstar. PUP/NIN is the uber solo when it can be used.
PUP/DNC is king when soloing monk class mobs, those guys will make /nin comopletely useless. Also, Animated Flourish is a good claiming ability.
Like I said, 99% of the time it's whm head, mage body, AKA the whm puppet (that's what your ls will learn to lovingly call it after you save their butts a few dozen times). The only time I solo with anything else is when I know I'm facing a mob that spams constant debuffs, then you put on what I've seen called the Cure Bot: valoredge head, mage body. The Cure Bot only casts cure spells.
The big ones are Mana Booster (reduces time between spells), Shock Absorber (after scorning it for so long, I have learned how amazing this attachment actually is), Mana Converter (deactivate/reactive is a fantastic way to set your activate timer to 20 minutes... which truly sucks), Mana Conserver (you only use Dark Maneuver to trigger the converter, so forget about the mana tanks), Damage Gauge (more likely to cast a healing spell), Stealth Screen (mage puppets cannot take a hit), Auto-repair I and II (regen is nice, aoe is everywhere, more hp on role reversal).
Avoid flashbulb, there is an internal timer on how often any effect attachment can fire off and you want to be able to use Shock Absorber and Mana Converter at will.
I don't have economizer, but if I did I would use it.
This is where I break with those who typically write guides. What I have put so far is an extremely defensive form of soloing, but it's not outside the realm of what people agree on as viable. IN MY OPINION the most important thing to do is to immediately pick up one merit in Role Reversal followed by one merit in Ventriloquy. Having these will send your survivability through the roof, you will literally wonder how you ever played before. Things that would have killed you will be a mere annoyance with Role Reversal. When soloing a mob you may lose hate over punching shadows and that would normally get your puppet destroyed... Ventriloquy to the rescue.
Honestly, after that it doesn't matter too much for soloing like this. Crit and hand-to-hand merits will make things go faster, but you never burn down a mob with this style of soloing as it is based around two premises: avoiding damage and recovering puppet mp faster than it is spent. If you want to get schnazzy you can merit evasion, but I don't have the combat merits to spend on that and I've never missed it.