Is it possible to make a class that has one hander light armor restoration? Pretty much a melee fighter that can also heal a dungeon.

Yes and no.

Light armour isn't the best for meleeing, but perked out will boost your magicka and magical abilities. You generally want a character to focus on either magicka or stamina, but you can respec fairly easily.

Templar or Dragonknight I'd recommend. Templar has access to useful burst heals, a class line dedicated to healing, Restoring Light. Dragonknight has an amazing self-heal, Green Dragon Blood, and some handy support abilities.

Most healing comes from either the Restoration Staff skill line or Templar's Restoring Light skill line. You DO NOT have to be a templar to be a healer, any class can fill any role. But if you're not using a resto staff, you'll probably need to be a templar.

At level 15, you unlock a second skill bar and weapon set to swap between. So you could have your sword and shield/dual-wield on one bar, and a staff on another. (A single one-handed sword is counted as 1h+shield)

For levelling, it doesn't really matter, but hybrid builds (focusing on magic and stamina equally) tend to be weaker than a build that prioritizes one over the other, but you could make it work. Melee dps uses stamina, healing mostly uses magicka, and they scale off of how much you have. Ultimate abilities scale off of the highest of those two stats.