This is an example of what we call a transferred epithet. Here's a definition from https://www.thoughtco.com/transferred-epithet-1692558: ....... A transferred epithet is a little known—but often used—figure of speech in which a modifier (usually an adjective) qualifies a noun other than the person or thing it is actually describing. In other words, the modifier or epithet is transferred from the noun it is meant to describe to another noun in the sentence. ......... In your sentence, it is the ploughman who is weary, but the adjective referring to him is transferred to another noun (his way). This serves to intensify the image by connecting his weariness to his journey. It conjures up a vivid image of weary worker trudging along with a slow and heavy step. You'll come across this device used in literature, but also sometimes in everyday life. For example, if we say "He had an unhappy marriage", this is a transferred epithet. 'Marriage' is a concept, so it is obviously not capable of feelings or emotions: marriage itself cannot be happy or unhappy. But we understand that the person was unhappy while he was married, just as we understand that the ploughman was tired as he was walking along.

It's a literary expression and is still current. [verb of motion e.g. go] + his weary way = [verb of motion e.g. go] + wearily

It's an expressive style and you can use both of them in formal and informal writing. "Weary" is kind of 'tired" but used for thing "way", which is very creative and interesting. There's a common say that : No sight is free of sadness. And "weary way" is a particular example.

Mainly in writing. In speaking, we usually say "tired".