The king can ask 1000 people to have a sip each, only the person who drank the poison will die after 4 weeks. Question does not limit the number of prisoners in any way.
Read the question carefully "Furthermore, it takes one month to have an effect"
1000 people drink one sip each out of 1000 bottles respectively.
1 month after this, 1/1000 people will die and you will know which bottle had poison.
I think it should be 1 and also there is a way which will include 1 prisoner into the experiment.
Hear me out.
We have 5 weeks remaining for the party i.e. 5*7 = 35 days and poison needs 30 or 29 or 31 days to take effect. So, we will simply take 1 prisoner and give him a sip of each bottle in interval of 10 mins. Note the timings. Afterwards note the time of his death also and by subtracting 1 month from death time we will get time at which he drank poisonous wine. Thus we got the poisonous bottle with 1 prisoner only.
Correct me if I am wrong.