No, Christianity did not copy ideas from other religions. Generally, the criticism is that there were religions that were older than Christianity that had similar concepts such as the Virgin birth, God becoming flesh, resurrection from the dead, etc. But, similarities don't mean that Christianity was copied from other religions any more than similarities in two different paintings mean that one was copied from another.
Christianity is derived largely out of the Old Testament Scriptures which contain the themes that are found in the New Testament. If one religion is older than another and, as the critics maintain, that which is new was borrowed from that which is older, then we would have to conclude that many religions borrowed from Judaism because the Old Testament, which was written between 400 and 1600 BC, was before them. Take a look at the following chart.
fulfilled in Jesus
|Ascension of Jesus to the right hand of God||Ps. 110:1||Matt 26:64; Acts 7:55-60; Eph. 1:20|
|Atonement by blood||Lev. 17:11||Heb. 9:22|
|Baptism||Exodus 40:12-15; Lev. 16:4; Gen. 17:10; Ezek. 36:25||Matt. 3:16; 28:19; Col. 2:11-12; Heb. 10:22|
|Begotten Son, Jesus is||Psalm 2:7||Acts 13:33; Heb. 1:5|
|Creative work||Gen. 1:26||John 1:1-3; Col. 1:16-17|
|Crucifixion||Psalm 22:11-18; Zech. 12:10||Luke 23:33-38|
|Damnation and Salvation||Dan. 12:2||Matt. 25:46|
|Eternal Son||Micah 5:1-2; Psalm 2:7||Heb. 1:5; 5:5|
|First and Last||Isaiah 41:4; 44:6; 48:12||Rev. 1:8,17; 22:13|
|God among His people||Isaiah 9:6; 40:3||John 1:1,14; 20:28; Col. 2:9; Matt. 3:3|
|Incarnation of God||Ex 3:14; Ps. 45:6 Isaiah 9:6; Zech. 12:10||John 8:58; 1:1,14; Heb. 1:8; Col. 2:9; Heb. 1:1-3|
|Monotheism||Isaiah 43:10; 44:6,8; 45:5||John 10:30; Eph. 4:5|
|Only Begotten Son||Gen. 22:2. See Typology||John 3:16; Heb. 11:7|
|Priesthood of Jesus||Psalm 110:4||Heb. 6:20; 7:25|
|Resurrection of Christ||Psalm 16:9-10; 49:15; Is. 26:19||John 2:19-21|
|Return of Christ||Zech. 14:1-5; Mic. 1:3-4||Matt. 16:27-28; Acts 1:11; 3:20|
|Sacrifice of the Son||Gen. 22. See Typology||Heb. 9:27|
|Salvation by grace||Gen. 12:3; Gen. 15:6; Hab. 2:4||Gal. 3:8-11; Rom. 4:9|
|Sin offering||Ex. 30:10; Lev. 4:3||Rom. 8:3; Heb. 10:18; 13:11|
|Sin offering made outside the camp||Ex. 29:14||Heb. 13:12-13|
|Sin offering without defect||Ex. 12:5; Lev. 22:20; Deut. 17:1||Heb. 9:14|
|Son of God||Psalm 2:7||John 5:18|
|Substitutionary Atonement||Isaiah 53:6-12; Lev. 6:4-10,21||Matt. 20:28; 1 Pet. 2:24; 2 Cor. 5:21; 1 Pet. 3:18;|
|Trinity||Gen. 1:1,26; Job 33:4; Gen. 17:1; 18:1; Ex. 6:2-3; 24:9-11; 33:20; Num. 12:6-8; Psalm 104:30; Gen. 19:24 with Amos 4:10-11; Is.48:16||John 1:1-3; John 1:18; 6:46; Matt. 28:19; 2 Cor. 13:14|
|Virgin Birth||Isaiah 7:14||Matt. 1:25|
|Worship of Jesus||Psalm 97:7||Matt. 2:2,11; 14:33; 28:9; John 9:35-38; Heb. 1:6|
As you can see from the above chart, Christianity is in agreement with the Old Testament Hebrew Scriptures. Many of the books of the Old Testament were written hundreds and hundreds of years before such religions as Mithraism. So, can we conclude that Mithraism borrowed from the Old Testament?
Furthermore, if the critics want to accuse Christianity of borrowing from pagan religions, then they need to establish their case. Simply making an assertion doesn't prove it is true. But since the Old Testament clearly shows New Testament themes we can conclude that Christianity did not copy from other religions. It fulfilled the Old Testament.