Before I start I wish to remind everyone:
While radiation in fiction usually is a strange mix of white and black magic that either make you a super-hero or turn you into a decomposing grotesquery, radiation in real life is not anything like that. Radiation is not different than heat, light, sound, wind or any other force of nature: too much of it will kill you, but it takes a LOT to get there.
Half your bullet list of how he would notice things are different do not apply in real life. No, animals and plants do not grow bigger in radiation. No, animals and plants do not deform or change that makes it any more noticeable beyond normal regional variation in species. If anything, stuff die. That is about the only thing your hero can notice at first glance; that some places appear sterile. The infamous Chernobyl Red Forest is the most classic case of this, and a very rare one at that.
And as far as getting Acute Radiation Sickness, well it would have to be some pretty severe radiation for that to happen. ARS kicks in at about 1000 mSv dose over a short time period (less than 24 hours). This can easily be the case for items, the Goiânia accident for instance demonstrated that. But the sphere of influence there was very localized. Once the source gets spread out over a large area, the intensity quickly diminishes. It is not impossible that an area can give ARS, but the nature of your apocalypse would play a big part here.
Anyway, the core of the question: assuming your hero knows basic physics, physiology and similar, would he discover radiation simply by noticing some items — or being in a specific area — makes him ill?
It is possible, yes.
The Star Trek: The Next Generation episode Thine Own Self shows a possible scenario with Data - suffering memory loss - ending up in a pre-industrial community, using basic science skills and some physics knowledge to deduce the existence of radiation. The radiation is coming from items in that case. It is a bit too long to write out the entire procedure; I recommend reading on the link.
Is this applicable to your scenario? Well it can be. Do note though that in your scenario, your hero would probably cut himself on Occam's Razor first. Since we are essentially talking about discovering what from his perspective actually is black magic — i.e. something completely unknown, invisible, without scent or sound, and that is causing ill health or even death — he is likely to assume other, simpler explanations first, like poisoning or infection. This is because symptoms alone are not enough to differ between ARS and ailments caused by pathogens or poisoning.
You have to let him rule out those things, while still maintaining enough curiosity to keep working with this black magic that is killing stuff and/or making him ill. It is not entirely impossible, like "Thine Own Self" showed, but you do have your work cut out for you. :)