Depolarisation of post synaptic neuron
When the neuron reaches threshold, this is the trigger that leads to sodium channels opening and this creates a positive feedback process causing the neuron to become less negative and more positive, triggering more sodium channels to open until it is inevitable an action potential will be reached. During an action potential, the neuron cell becomes more positive compared to the outside for a very short period(ie, 1 millisecond or less). 

Yes, depolarisation does mean the inside becomes less negative in this context.
You would expect the inside of the neuron to be at +55mV (or perhaps more) during an action potential, as the depolarisation of the neuron grows closer to reaching the sodium equilibrium potential which is estimated to be around +58mV. However, during the phase leading up to an action potential, potassium channels also remain open. The diffusion of potassium ions can lead to the cell becoming more negative. In other words, it limits the voltage of an action potential to around +33mV instead of +55mV.

Sources:

(1)https://www.khanacademy.org/science/biology/human-biology/neuron-nervous-system/a/depolarization-hyperpolarization-and-action-potentials 

(2)https://opentextbc.ca/anatomyandphysiology/chapter/12-4-the-action-potential/