Various media reports indicate that Johnson's situation is currently critical.
He has undergone surgery, though no diagnosis has been publicly released other than it's supposedly not a heart attack or stroke.
Update: An ER doctor blogger speculates given the evidence at hand.
Given the information at hand, I suspect that the Senator had either a TIA or possibly a mild stroke, and underwent carotid endarterectomy. This would imply that his illness might be mild enough to allow him to serve out his term, at the least, and make a full recovery, at the best. It is always possible, and alas, too common, that early media report are inaccurate or incomplete, so I would not hold this estimate out as definitive.
He explains "TIA" as transient ischemia.
"[He] was able to walk back to his office in the Hart Senate Office Building, then began having problems with his right arm. He thought he was all right, she said, and went to his desk, but came out a few minutes later and it was apparent he needed help." The language symptoms having resolved, he developed new symptoms of a motor nature. The language centers (often called "Broca's area") are typically in the left brain, in the same vascular distribution as the left motor cortex, which controls movement of the right arm. Again, this is consistent with cerebral ischemia, though the intermittent nature of the symptoms and variable type of symptomology suggest transient ischemia (often called TIA or mini-stroke) versus true cerebral infarction, aka stroke.
As for the surgery:
What surgery? Don't know as of this moment. But if he had a TIA, that would generally involve a Carotid Ultrasound as part of the ER work-up. If that revealed a critical stenosis of the carotid artery, it is possible that he would proceed emergently to the operating room to open the offending artery via a procedure called carotid endarterectomy [...] Generally, carotid endarterectomies are not performed emergently, but in the case of a senator, VIP medicine might well motivate a vascular surgeon to do so.