You could perform string slicing and concatenation provided that your dataset comes in a predictable and standard format. Cast this new string to datetime using the pd.to_datetime method.
For example, this would work for your example:
import pandas as pd
df = pd.DataFrame([[13, 'Septiembre /98'], [15, 'August/98'], [24, 'Novem /98']], columns=["Day", "Month and year"])
df['Date'] = pd.to_datetime(
    df['Day'].astype('str') +
    ' - ' +
    df['Month and year'].str.slice(0, 3) +
    ' - ' +
    df['Month and year'].str.slice(-2)
)
print(df)
   Day  Month and year       Date
0   13  Septiembre /98 1998-09-13
1   15       August/98 1998-08-15
2   24       Novem /98 1998-11-24