If "closest star" is Proxima Centauri, we're talking 4.5 light-years. Getting there in 1000 years means an average speed of 0.45% of the speed of light. At that speed less than 0.01 cm of titanium is enough shielding to keep the radiation out according to figure 1c) in my source below. Which makes sense because because this is ~10% of the speed of "normal" alpha radiation which is stopped by just your skin.
"Radiation Hazard of Relativistic Interstellar Flight" by Oleg Semyonov: https://arxiv.org/abs/physics/0610030 via Project Rho: https://projectrho.com/public_html/rocket/slowerlight3.php