Modeling the popularity dynamics of an online item is an important open problem in computational social science. This paper presents an in-depth study of popularity dynamics under external promotions, especially in predicting popularity jumps of online videos, and determining effective and efficient schedules to promote online content. The recently proposed Hawkes Intensity Process (HIP) models popularity as a non-linear interplay between exogenous stimuli and the endogenous reactions. Here, we propose two novel metrics based on HIP: to describe popularity gain per unit of promotion, and to quantify the time it takes for such effects to unfold. We make increasingly accurate forecasts of future popularity by including information about the intrinsic properties of the video, promotions it receives, and the non-linear effects of popularity ranking. We illustrate by simulation the interplay between the unfolding of popularity over time, and the time-sensitive value of resources. Lastly, our model lends a novel explanation of the commonly adopted periodic and constant promotion strategy in advertising, as increasing the perceived viral potential. This study provides quantitative guidelines about setting promotion schedules considering content virality, timing, and economics.