Thanks for your question.
Is suicide sinful? Suicide is self-murder and it is most definitely sinful. By committing suicide, a person is saying, "My problems are too big for God to help me solve." This is a lack of faith. Whatever denies faith is sin. See Romans 14:23 "...for whatever is not from faith is sin."
No problem is bigger than God. Hear what the apostle John says in 1 John 5:4 "For everyone who has been born of God overcomes the world. And this is the victory that has overcome the world––our faith." This is what Paul says in Philippians 4:13 "I can do all things through Christ who strengthens me." If we have faith, we can overcome every obstacle in our path; defeat any temptation; achieve any goal. To commit suicide is to say that this is not true and that God is smaller than our problems and that He doesn't care enough to help us in time of need. Salvation is impossible for one with such an attitude. However, if we will take the time to build a real faith - one that is built solidly on His truth as revealed in His word, then, as Paul says in Romans 8:37, we will be more than conquerors in Christ. Romans 8:37-39 "Yet in all these things we are more than conquerors through Him who loved us. For I am persuaded that neither death nor life, nor angels nor principalities nor powers, nor things present nor things to come, nor height nor depth, nor any other created thing, shall be able to separate us from the love of God which is in Christ Jesus our Lord."
There may be some exceptions to the rule that suicide is sin. Some people are beset by a medical/mental condition which leads them to take their own lives. They may not be responsible for their actions. We know that God judges righteously and will not punish a person for doing something over which he or she has no control. Such a person would do well to take advantage of the wonderful advances in medical and psychological science available to us today to help in dealing with such conditions. These advances are blessings from God and we would be foolish not to make use of them when the situation calls for such intervention.
There are two other aspects to your question. Firstly, you are really asking, is there salvation for anyone if they are not "in Christ"? The answer to that is no. If there is, then the Bible is untrue. See Acts 4:12; 10:43; Mark 16:15,16; John 3:36; John 14:6. So, if a person is not "in Christ" he or she cannot have their sins forgiven whatever they do or however they behave in life. Secondly, I think you are probably fearful of the terrible false doctrine in which the "unforgiven" are sent to a burning hell to be tortured forever. When you think about it this is a most horrifying doctrine and absolutely blasphemous because it portrays God as being vengeful and vindictive beyond the wildest dreams of even the host horrible of this world's tyrants and war criminals. The Bible knows nothing of such a vile doctrine. Rather the issue is a stark one: life or death. See Deuteronomy 30:19; Romans 6:23. When someone who is not "in Christ" dies, whether through natural causes, violence, or suicide, that is the end of them. They are extinguished but they are NOT tortured forever in the flames.
If you want to come to grips with what the Bible really teaches, why not take the free online Bible study course available on our website, https://www.thisisyourbible.com? This course will give you a background in the major themes of the Bible from Genesis to Revelation. As with everything we offer, there is no cost to you other than your time and effort. You will also have a personal tutor to whom you may pose questions either from the course itself or those questions which come to you as you read the Bible.
I hope you have found this helpful.God bless!