If a spellcaster casts haste on a monster with Multiattack, does he/she get to make the extra attack (one weapon attack only) from haste after his/her Multiattack?
Yes, a hasted monster can Attack and Multiattack in the same turn
It gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
Multiattack is a special action that monsters can take, it is listed in the Monster Manual (p. 11) under the Actions section.
If a monster takes the Multiattack action, then they can still use the haste action for any of the actions allowed by it, including to attack. In this way the monster could Multiattack and Attack in the same turn while hasted.
It is worth noting that the order in which you take these actions is up to you so you may take the extra Attack action from haste before the Multiattack, for example.
Note: a hasted monster cannot Multiattack twice using the haste action
A related question to the one you seem to have asked (but included for completeness in case you were curious) is if one can use the extra action granted by haste to take another Multiattack action. To which the answer is no.
Multiattack is a special action that monsters can take, but it is not any of the actions listed in haste.
Jeremy Crawford has clarified that:
Multiattack and Attack are different actions.
Attack is listed in haste as an acceptable action, but Multiattack is not.
Therefore, a monster cannot use the additional action granted by haste to take another Multiattack.
See this question for more on that aspect: If a creature with Multiattack is affected by Haste, can it make two Multiattacks?