Blood rush is an activated ability, therefore it is instant speed.
Unless stated otherwise, all activated abilities are instant-speed.
Conceptually, it would have to be instant-speed. As things cast at sorcery-speed can only be done during a main phase, it would be unable to actually do anything for the creatures, as they are no longer attacking when post-combat main phase occurs.
So, if you're attacking with a creature, and you go through declare blockers stage, and there's no blocker declared for a certain creature, can you blood rush that creature aftter the declare blockers stage to deal tons of damage directly to them?
Yes, you could. There is always a round of priority before the game moves onto the next phase; this includes any phase of combat (beginning of combat, declare attackers, declare blockers, combat damage, end of combat).
I have my naya deck playing against a jund deck. I'm atking with my creature 3/3 and my opponent declares no blockers on the atking creatures and asked me "I'm not blocking, are you dealing the damage now?" I have a ghor-clan rampager in my hand and enough mana to bloodrush it but I'm not going to waste it unless he is blocking so I said "yes, I'm dealing 3 damage to you" and then he casted the searing spear on my 3/3. And then I said "in response, I'll bloodrush 4/4 and trample into my 3/3" but he said the question he asked makes himself to have the priority and I cannot do the bloodrush afterwards. Is that true??
You certainly can Bloodrush while the Searing Spear is on the stack. Nothing resolves on the stack or moves to the next phase of the turn without 'both' players passing priority.
Again, Bloodrush is an activated ability; and unless otherwise stated in the effect, all activated abilities are instant-speed.
Greatsword is correct, you always have the option of responding and can use the blood rush ability at instant speed to Prevent the searing spears from killing your creature.