How can I "merge" rows by same value in a column in Pandas with aggregation functions?

You are looking for

aggregation_functions = {'price': 'sum', 'amount': 'sum', 'name': 'first'}
df_new = df.groupby(df['id']).aggregate(aggregation_functions)

which gives

    price     name  amount
id                        
1     130     anna       3
2      42      bob      30
3       3  charlie     110

For same columns ordering is necessary add reindex, because aggregate by dict:

d = {'price': 'sum', 'name': 'first', 'amount': 'sum'}
df_new = df.groupby('id', as_index=False).aggregate(d).reindex(columns=df.columns)
print (df_new)
   amount  id     name  price
0       3   1     anna    130
1      30   2      bob     42
2     110   3  charlie      3