The computer simulations of fluctuational dynamics of an annular system governed by the sine-Gordon model with a white noise source are performed. It is demonstrated that the mean escape time (MET) of a phase string for an annular structure can be much larger than for a linear one and has a strongly pronounced maximum as a function of system's length. The location of the MET maximum roughly equals the size of the kink-antikink pair, which leads to evidence of a spatial crossover between two dynamical regimes: when the phase string escapes over the potential barrier as a whole and when the creation of kink-antikink pairs is the main mechanism of the escape process. For large lengths and in the limit of small noise intensity gamma, for both MET and inverse concentration of kinks, we observe the same dependence versus the kink energy E(k): approximately exp(2E(k)/gamma) for the annular structure and approximately exp(E(k)/gamma) for the linear one.