This is a very well-known and much-loved American Christian hymn, often sung at funerals, but at other gatherings as well, formal and informal. It is sung at services of every Christian denomination I know of, Catholic and Protestant both.
1) Yes, "He will my shield and portion be" has the same meaning as "He will be my shield and portion." The former is a poetic (and archaic, I think) phrasing, used rhymically and to make the lines rhyme; you wouldn't speak this way.
2) "Portion" here is in the sense of the singer's lot in life. The word is most commonly used now to refer to a serving of food, as in "a six-ounce portion of beef." I think the song's usage is uncommon now, except for someone speaking in an "old-fashioned" way for effect: "I suppose it's just my portion in life to wind up with lousy jobs and loser boyfriends," Cynthia complained to her best friend.
3) "...within the veil..." I have never heard this verse before, but I believe it means in heaven, after death. When people die they are sometimes said to "pass beyond the veil." The metaphor is taken from the veil in the Temple at Jerusalem, which separated the holy area, where only the high priest could go, from the rest of the sanctuary.
4) "No less days to sing God's praise then when we first begun." Well, it's not very grammatical--should be "when we began"---but this refers to our eternal life in heaven. After ten thousand years, there will literally be as much time left (an infinite amount) as when we first arrived. Mathematicians, please correct if this is wrong :)
You don't seem to be asking about "wretch" in this posting, but in this hymn the word has the meaning of "a pitiable person." This word has shifted somewhat in meaning (and in usage; you don't hear it very often, at least not in AE) since the hymn was written. Some churches (Catholic, IIRC) subsitute the lyric "Amazing grace! How sweet the sound/ That saved and set me free..."
[not a teacher]