How to pass another entire column as argument to pandas fillna()

You can provide this column to fillna (see docs), it will use those values on matching indexes to fill:

In [17]: df['Cat1'].fillna(df['Cat2'])
Out[17]:
0    cat
1    dog
2    cat
3    ant
Name: Cat1, dtype: object

You could do

df.Cat1 = np.where(df.Cat1.isnull(), df.Cat2, df.Cat1)

The overall construct on the RHS uses the ternary pattern from the pandas cookbook (which it pays to read in any case). It's a vector version of a? b: c.

Just use the value parameter instead of method:

In [20]: df
Out[20]:
  Cat1      Cat2  Day
0  cat     mouse    1
1  dog  elephant    2
2  cat     giraf    3
3  NaN       ant    4

In [21]: df.Cat1 = df.Cat1.fillna(value=df.Cat2)

In [22]: df
Out[22]:
  Cat1      Cat2  Day
0  cat     mouse    1
1  dog  elephant    2
2  cat     giraf    3
3  ant       ant    4

pandas.DataFrame.combine_first also works.

(Attention: since "Result index columns will be the union of the respective indexes and columns", you should check the index and columns are matched.)

import numpy as np
import pandas as pd
df = pd.DataFrame([["1","cat","mouse"],
    ["2","dog","elephant"],
    ["3","cat","giraf"],
    ["4",np.nan,"ant"]],columns=["Day","Cat1","Cat2"])

In: df["Cat1"].combine_first(df["Cat2"])
Out: 
0    cat
1    dog
2    cat
3    ant
Name: Cat1, dtype: object

Compare with other answers:

%timeit df["Cat1"].combine_first(df["Cat2"])
181 µs ± 11.3 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

%timeit df['Cat1'].fillna(df['Cat2'])
253 µs ± 10.3 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

%timeit np.where(df.Cat1.isnull(), df.Cat2, df.Cat1)
88.1 µs ± 793 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)

I didn't use this method below:

def is_missing(Cat1,Cat2):    
    if np.isnan(Cat1):        
        return Cat2
    else:
        return Cat1

df['Cat1'] = df.apply(lambda x: is_missing(x['Cat1'],x['Cat2']),axis=1)

because it will raise an Exception:

TypeError: ("ufunc 'isnan' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''", 'occurred at index 0')

which means np.isnan can be applied to NumPy arrays of native dtype (such as np.float64), but raises TypeError when applied to object arrays.

So I revise the method:

def is_missing(Cat1,Cat2):    
    if pd.isnull(Cat1):        
        return Cat2
    else:
        return Cat1

%timeit df.apply(lambda x: is_missing(x['Cat1'],x['Cat2']),axis=1)
701 µs ± 7.38 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)