Hypertension is one of the most common diseases in humans, and there is a special concern on the consequences of maternal hypertensive conditions for the health of newborns. An inherited stress-induced arterial hypertension (ISIAH) rat strain has been selected but only a few studies have addressed behavior in these rats. Body weight, neurodevelopmental reflexes, and neuronal density in the hippocampus were compared in ISIAH and normotensive WAG rats during their suckling period. Systolic and diastolic blood pressure (SBP, DBP), adult rat performance in the open field (OF), elevated plus maze (EPM), and novel object recognition (NOR) tests were evaluated at the age of 12–14 weeks old. Body weight in pups did not differ significantly during the suckling period, while adult ISIAH rats were heavier than age-matched WAG rats and possessed the increased SBP and DBP. ISIAH pups were developmentally more advanced than WAG as indicated by grasp reflex and negative geotaxis reaction scores. This was associated with higher neuronal density in CA1 and CA3 hippocampal areas in ISIAH pups on postnatal day 6 as compared to WAG rats. Adult ISIAH rats demonstrated an increased locomotor and exploratory activity in the OF and EPM tests as well as low levels of anxiety. The NOR test revealed no significant difference in recognition but confirmed higher exploratory activity in ISIAH rats compared to WAG rats. The results indicate that hypertensive ISIAH rats feature accelerated development during their suckling period, and as adults, they are more active and less anxious than normotensive WAG rats.