How should Christians apply promises that God gave to the Israelites?
There are two things I want to address here: first your question, and second your interpretation of Proverbs 3 :1-2.
Your initial question is an excellent one and should be contemplated by all believers who take the time to read and study the Bible, especially the Old Testament. There is far too great a tendency for Christians to assume that everything we read is for our personal application. While the Holy Spirit is free to apply anything to anyone, we need to understand that there are many things in the Old Testament that were exclusively intended for Israel. And the majority of those are included within the expressed terms of the Mosaic Covenant that God made with Israel.
God made many promises to Israel which were conditioned upon their obedience to the commands of the Mosaic covenant. His main promise was to bless them in the land. And that included all kinds of wonderful promises, not the least of which was perpetual success against their enemies, fruitfulness both physically and agriculturally and even long life.
The question is...are these promises something that Christians can claim today?
The answer is no. And the reason is two-fold:
1. The Church does NOT — as some have wrongfully taught — take the place of Israel, and…
2. Because we, the Body of Christ, do not function under the terms of the Mosaic Covenant.
Ours is a covenant of grace, the terms of which were fulfilled by Christ. And the promises made to us are entirely different than those extended to Israel. Rather than a physical kingdom with physical blessings, we have been offered “better promises” — according to Hebrews 8:6. These include a spiritual kingdom with spiritual blessings. (See Ephesians 1:3)
For further insight, I would suggest you listen to my two studies through Hebrews chapter 8.
Hebrews 8:1-7 and Hebrews 8:8-13
Concerning your reference to Proverbs 3:1-2, which says:
My son, do not forget my teaching, but let your heart keep my commandments, for length of days and years of life and peace they will add to you. (ESV)
I agree this passage looks very much like a promise from God, but we need to remember that Proverbs is a book of wisdom. As such, it challenges the reader to live a life of wisdom, and then offers a generalized observation of what one can expect for doing so. In other words, this passage is saying that, generally speaking, those who are subject to their parents live longer and better lives. Conversely, those who resist the discipline of their parents and authority in general generally invite all kinds of negative results, such as sickness, accidents, tragedies and even premature death.
A wise and articulate fellow Calvary Chapel pastor once said, "Proverbs are principals drawn from observation, not promises given by revelation."
God bless you!