Can a Jew go to heaven?
We use the word "Jew" in two different senses. A person can be a Jew because of his ancestry; that is, he descends from Jewish people. A person can also be a Jew because that is the religion he practices.
As an ethnic group, Jews can be saved just like any other nation. "For you are all sons of God through faith in Christ Jesus. For as many of you as were baptized into Christ have put on Christ. There is neither Jew nor Greek, there is neither slave nor free, there is neither male nor female; for you are all one in Christ Jesus. And if you are Christ's, then you are Abraham's seed, and heirs according to the promise" (Galatians 3:26-29). Everyone is offered the ability to be saved by the same set of rules. There isn't one set of rules for the Jews and another for the Gentiles, though this is popular teaching among groups who call themselves "Messianic Jews."
So, along with this, no one can be saved today by following the Old Testament laws. "Stand fast therefore in the liberty by which Christ has made us free, and do not be entangled again with a yoke of bondage. Indeed I, Paul, say to you that if you become circumcised, Christ will profit you nothing. And I testify again to every man who becomes circumcised that he is a debtor to keep the whole law. You have become estranged from Christ, you who attempt to be justified by law; you have fallen from grace" (Galatians 5:1-4). The Mosaical law taught the world what was sin, but it could not offer a remedy for sin -- other than to point to the coming Messiah.
"For the law, having a shadow of the good things to come, and not the very image of the things, can never with these same sacrifices, which they offer continually year by year, make those who approach perfect. For then would they not have ceased to be offered? For the worshipers, once purified, would have had no more consciousness of sins. But in those sacrifices there is a reminder of sins every year. For it is not possible that the blood of bulls and goats could take away sins" (Hebrews 10:1-4).
"Therefore, my brethren, you also have become dead to the law through the body of Christ, that you may be married to another--to Him who was raised from the dead, that we should bear fruit to God. For when we were in the flesh, the sinful passions which were aroused by the law were at work in our members to bear fruit to death. But now we have been delivered from the law, having died to what we were held by, so that we should serve in the newness of the Spirit and not in the oldness of the letter" (Romans 7:4-6).
Those who lived before Christ's death did gain salvation, though they lived under the Mosaical Law. That is because Christ didn't just die for the future, he also died for the past. "And for this reason He is the Mediator of the new covenant, by means of death, for the redemption of the transgressions under the first covenant, that those who are called may receive the promise of the eternal inheritance" (Hebrews 9:15).
So in short, a Jew cannot be saved by following Jewish law today. He can only be saved by becoming a Christian, just like all others. "Nor is there salvation in any other, for there is no other name under heaven given among men by which we must be saved" (Acts 4:12).