At the end of the ability, it states that in all other ways it behaves like a normal Barbarian Rage.
So I would imagine, no, you can't use Spells, Supernatural or Spell-like Abilities.
You can use SU abilities while in a rage. SU abilities explicitly don't require concentration and only require patience if the DM interprets them like that. This came up a while ago talking about raging bards. Their perform ability (which is SU) seems doable while in a rage (unless you think 6 seconds of performance requires patience).
A barbarian can call upon inner reserves of strength and ferocity, granting her additional combat prowess. Starting at 1st level, a barbarian can rage for a number of rounds per day equal to 4 + her Constitution modifier. At each level after 1st, she can rage for 2 additional rounds. Temporary increases to Constitution, such as those gained from rage and spells like bear's endurance, do not increase the total number of rounds that a barbarian can rage per day. A barbarian can enter rage as a free action. The total number of rounds of rage per day is renewed after resting for 8 hours, although these hours do not need to be consecutive.
While in rage, a barbarian gains a +4 morale bonus to her Strength and Constitution, as well as a +2 morale bonus on Will saves. In addition, she takes a –2 penalty to Armor Class. The increase to Constitution grants the barbarian 2 hit points per Hit Dice, but these disappear when the rage ends and are not lost first like temporary hit points. While in rage, a barbarian cannot use any Charisma-, Dexterity-, or Intelligence-based skills (except Acrobatics, Fly, Intimidate, and Ride) or any ability that requires patience or concentration.
A barbarian can end her rage as a free action and is fatigued after rage for a number of rounds equal to 2 times the number of rounds spent in the rage. A barbarian cannot enter a new rage while fatigued or exhausted but can otherwise enter rage multiple times during a single encounter or combat. If a barbarian falls unconscious, her rage immediately ends, placing her in peril of death.
Supernatural Abilities (Su): Using a supernatural ability is usually a standard action (unless defined otherwise by the ability's description). Its use cannot be disrupted, does not require concentration, and does not provoke attacks of opportunity.
I personally think it was an oversight and controlled rage should allow the stuff that requires patience and concentration again.
But I don't think by RAW it does.
Anyway, I'm FAQing this, maybe we get a Dev to chime in.
Oh, and to respond to OP, by RAW no, you cannot use anything that requires patience or concentration. That part of rage is not modified by the Urban Barbarian archetype.
I think this is RAI and not an oversight, but that is just an opinion.
...Right, I was even in that thread. "Slip of the tongue" I should say, then.
Hexes are what? Supernatural? I mean, I suppose I can see it. I'm not exactly the Allmighty Arbiter of Law, so take my opinion with a grain of salt.
Almost all (if not all, I can't find any in my 3 second search that are not) are SU. So it just comes down to if the DM thinks any given hex requires patience.
I believe a few specifically use the term 'concentrate' or 'concentration' and thus could not be used in a rage, but they are in the minority.
@Rycaut: Also, one should not use the word 'cast' in conjunction with SU! It creates confusion. SUs are not 'cast' they are used.
Not by RAW. The ubarbarian can only end the rage and can start a new one with different bonuses.