In this book, unless otherwise noted, the word "our" refers to Paul and Timothy (the one to whom this letter is written), as well as to all believers. (See: [[rc://en/ta/man/jit/figs-exclusive]])
"I, Paul, wrote this letter. I am an apostle." Your language may have a particular way of introducing the author of a letter. Immediately after introducing the writer, you may want to indicate to whom the letter was written, as in the UDB.
Here "our hope" refers to the person in whom we have confidence. Alternate translation: "Christ Jesus, who is the one in whom we have confidence" or "Christ Jesus, whom we trust"