The word "tears" eventually rhymes with "years." This is neat because it makes us think the tears will last for many "years," which is kind of what the speaker ends up telling us later in the poem.
Half broken-hearted To sever for years, (3-4)
Why "half broken-hearted"? Something tells us that maybe the sadness was too bad. After all, if this moment was so bad, wouldn't they have been completely broken-hearted, or totally broken-hearted?
Truly that hour foretold Sorrow to this. (7-8)
The speaker, apparently, totally knew he would be sad. The miserable "hour" when they said goodbye pretty much predicted his current "sorrow." Maybe he's looking at it all wrong.
The dew of the morning Sunk chill on my brow— It felt like the warning Of what I feel now. (9-12)
The idea of prediction or foreshadowing is picked up again. The chilly dew on his brow was a dead giveaway that he would feel really sad. Couldn't he have interpreted the "hour" and "dew" differently and pretended they didn't foretell anything? Just an idea.
A shudder comes o'er me— Why wert thou so dear? (19-20)
We shudder when we're scared but also when we're upset. Clearly the speaker is sad and frustrated when he hears the woman's name. But his sadness and irritation stem from the fact that this woman meant so much to him. It's almost like he says "if only I didn't like you so much."
Long, long shall I rue thee, Too deeply to tell. (23-24)
"Rue" is a really bizarre word to use here. Usually, we "rue" something we regret because it has a bunch of consequences we don't like (like that huge stomach ache you'll surely get if you eat an entire extra large pizza by yourself). Well, seeing as how that's the case, the speaker seems less sad and more regretful or annoyed. The plot thickens…
In silence I grieve, That thy heart could forget, Thy spirit deceive (26-28)
The speaker seems more upset that the woman has totally forgotten about him—her spirit is misleading her. It's not that he doesn't have her. It's that she has another man. Hmm, so he's sad about a few things.
How should I greet thee?— With silence and tears (31-32)
The speaker's sadness seems eternal. If he and his friend meet again, there won't be any rejoicing or "OMG it's so good to see you!" Their reunion will be like their parting: silent, tearful, and sad. Well, that's depressing. Sheesh.