So reading the description of Spell Rage, it would seem that it allows for all spell casting to be used while in Rage correct? It does state you need to be an arcane caster to take the PrC but not that those are the only spells able to be cast while in a Spell Rage. I'm trying to make an Obad-Hai Cleric Barbarian.
You are correct that the spell rage feature makes no distinction between arcane and divine spellcasting, and works equally well for both. If a barbarian cleric can enter the class, the cleric spellcasting benefits from spell rage (but not from the even-level spellcasting advancement, which does specify arcane).
But what spell rage actually does for spellcasting, be it arcane or divine or something else (an artificer’s infusions maybe?) is much less clear. The class feature rules text bounces back and forth between referring to “spell rage” and “rage” and it is not necessarily clear what, if any, of the class feature should apply to regular rage as well as spell rage. See this related question for more on this confusion.
I believe the most likely thing is that you only get to cast spells during a spell rage (which could be at the same time as a regular rage but wouldn’t have to be), which is a problem because you only get that 1/day at 1st and can only ever get at most 2/day total. There is no way to get more, as you can with rage via Extra Rage or other options.
I would talk to your DM about this. You could ask for a divine variant to the rage mage to ease your entry, you could ask for improvements to its spellcasting so it isn’t such a trap, and you could ask for clarity on what spell rage does and for ways to ensure you can actually get enough of it to be useful. In fact, you might ask for this homebrew rage sage class instead, which is much better and I heartily recommend (despite the way forum updates have disrupted the table there).