So, the Spellslinger is all kinds of cool, in theory. I'm not sure straight Spellslinger is worth it, though: Everything you need comes up front, and levels 2-20 are just nerfed wizard levels.
Rules issues:
1) Can spells from other classes be used to power Mage Bullets? Sure. The FAQ for Bloodline Arcana covers Mage Bullets too.
2) Can spell slots from a spontaneous caster be used to power Mage Bullets? I'd say so. The ability refers to sacrificing "a spell", not "a prepared spell". Usually, I'm not the one arguing semantics like this, but spell/prepared spell/spell slot is a distinction that the writers and editors are usually clear on, and it works fine with the flavor, so I see no reason why it wouldn't work.
So, ideas:
*Myrmidarch Magus. There's thematic overlap, but then I remember that Ranged Spellstrike is only useful for extending the range of short-ranged touch spells by using dye arrows and tangle bolts.
*Regular Magus. You can use a swordcane-pistol; stab a guy, step back, cast scorching ray through your enhanced gun...you'll retain the use of Spell Combat and be a switch hitter, but you're not doing anything interesting with Spellstrike, and Mage Bullets won't help your melee attacks.
*Bladebound. If you can convince your GM the the black blade swordcane-pistol you're bound to gets the enhancement bonus for both blade AND barrel, then you can probably convince him of something far more useful.
*Cleric or Druid. Good casting progression. Wearing armor doesn't hinder your spellslinging.
*Sorcerer. Best blasting spells, and you're basically giving up 1 level of sorcerer for a scaling bonus to most of the blasting spells.
*Eldritch Knight (aasimar or drow for early entry). Fighter 1/Spellslinger 1/EK. Or Spellslinger/sorcerer/fighter/EK. Or replace Fighter with Magus; the ability to add another +1 to your gun might be worth +1 BaB and a feat.
*Mystic Theurge. Spellslinger 1/Cleric 2/MT (with requirements met by racial SLAs and domain abilities).
Other ideas?
I recommend discussing the mage bullets rules with your GM to make sure you are both reading them the same way. Eg half of the weapon abilities can only apply to melee weapons yet the rule as written states that you can only apply them to a single barrel of your gun - go through each one in turn. Plus he/she might allow you to sacrifice more than one spell at a time if you weasel enough.
If you're going to be the primary spellcaster then you might be better being a normal wizard and take a level of gunslinger to get the gun abilities, followed by eldritch knight.
I'm currently playing a spellslinger 1/magus X with which the DM let me have the Ranged Spellstrike ability for a plot-related character disadvantage. Gimmick-wise, I'm focusing on the Snowball spell while stacking the spellslinger's Mage Bullets and the magus' Arcane Pool to enhance the gun (and thereby also the save DC for Snowball's staggering effect via the spellslinger's Arcane Gun). It's not that a myrmidarch couldn't also do this, but the Spell Recall ability makes it much easier to use as a staple while also having some other, more interesting spells memorised as well.
I will say, however, that it's been difficult to find magus spells that work well with this angle, at least at lower levels. You'd think Scorching Ray would be the go-to spell for this, but you can spellstrike with more than 1 ray until level 11, as I read the ability, so Snowball is often the better choice. I like the idea of using Hydraulic Push through the gun, but the mechanics work in such a way that it's not augmented by the spellslinger's abilities.
Gust of Wind would benefit, but I struggle to find uses for it.