This is the common prayer that we hear whenever Christians pray! They pray generally and not specifically. The question is, "Should we Christians during this present dispensation of Grace or during this New Testament time, pray to God that He will bless us?" I personally don't see any teaching in the New testament even in Apostle Paul's letters (who is our apostle, we gentiles), that teaches us to pray "God bless me". Instead, we can read in Ephesians 1:15-19
"Wherefore I also, after I heard of your faith in the Lord Jesus, and love unto all the saints, Cease not to give thanks for you, making mention of you in my prayers; That the God of our Lord Jesus Christ, the Father of glory, may give unto you the spirit of wisdom and revelation in the knowledge of him: The eyes of your understanding being enlightened; that ye may know what is the hope of his calling, and what the riches of the glory of his inheritance in the saints, And what is the exceeding greatness of his power to us-ward who believe, according to the working of his mighty power..."
Pauls's prayer was not "Lord bless them", but the specific prayer "God, give them spirit of wisdom and revelation". That's the Biblical and apostolic prayer.
And also, I believe we don't have to pray "Lord, bless me", because that prayer during this dispensation is a prayer of doubt and unbelief. Why? Because Ephesians 1:3 teaches "Blessed be the God and Father of our Lord Jesus Christ, who hath blessed us with all spiritual blessings in heavenly places in Christ". It says, God HAS already BLESSED us, not He will bless us. I am blessed, you are blessed, we are blessed. Why ask God to bless you, if He has already blessed you. It's better to pray like this, "Lord, thank You for You have already blessed me!". We are already blessed. Receive it by faith. Don't doubt.
You may say, "hey, brother...Ephesians 1:3 says we are blessed with SPIRITUAL blessings, and not MATERIAL blessings". Yes, in fact it is "SPIRITUAL". But spiritual doesn't mean "not real" or just "psychological", but spiritual is a fact, it is truth. Proverbs teaches in Proverbs 10:22 "The blessing of the LORD, it maketh rich, and he addeth no sorrow with it." The blessing of the LORD which is "spiritual" maketh rich