The poem's predominant rhythm is iambic tetrameter. This means that most lines have four feet, each foot consisting of one unstressed (or unaccented) syllable followed by one stressed (or accented) syllable. Of course, there are a couple of lines that are exceptions to this. I have placed accented syllables in bold in the poem below:
My heart | leaps up | when I | be hold (iambic tetrameter)
A rain | bow in | the sky (iambic trimeter)
So was | it when | my life | be gan (iambic tetrameter)
So is | it now | I am | a man (iambic tetrameter)
So be | it when | I shall | grow old (iambic tetrameter)
Or let | me die (iambic dimeter, or one iamb and one spondee)
The Child | is fa | ther of | the Man (iambic tetrameter)
And I | could wish | my days | to be (iambic tetrameter)
Bound each | to each | by nat' | ral pi | e ty (iambic pentameter)
Lines 2, 6, and 9 are irregular and do not follow the regular iambic tetrameter of the rest of the poem. Line 2 is in iambic trimeter (which means there are three feet, not four). Line 6 only has two feet, and this is called dimeter: the first foot is an iamb and the second is a spondee (where both syllables are accented or stressed). The final line is the toughest, I think, to scan. ("Scanning" is the process of marking the stressed syllables in a line of verse.) When we say the word "natural," we often make it two syllables (as in "natch-rull") instead of three syllables (as in "natch-a-rull"); therefore, if we split it only into two syllables rather than three, the line scans neatly as iambic pentameter (which has five feet instead of four). However, because six of the nine lines employ regular iambic tetrameter, we can name this as the poem's rhythm.
The rhyme scheme is abccabcdd. This means that lines 1 and 5 rhyme (a), lines 2 and 6 rhyme (b), lines 3, 4, and 7 rhyme (c), and lines 8 and 9 rhyme (d).