Yes, there are the string justification methods, ljust and rjust.

>>> '12'.rjust(5, '#')
'###12'
>>> 'txt'.rjust(5, ' ')
'  txt'
>>> '12'.ljust(5, '#')
'12###'

As maybe a alternative more portable [1] and efficient [2], actually you can just use str.ljust.

In [2]: '190'.ljust(8, '0')
Out[2]: '19000000'

In [3]: str.ljust?
Docstring:
S.ljust(width[, fillchar]) -> str

Return S left-justified in a Unicode string of length width. Padding is
done using the specified fill character (default is a space).
Type:      method_descriptor

[1] format is not present on old python versions. format specifier was added since Python 3.0 (see PEP 3101) and Python 2.6.

[2] reverse twice is an expensive operation.

Let us do np.where combine with zfill, alternative you can check with str.pad

df.Random=np.where(df.Random.str.len()<9,df.Random.str.zfill(9),df.Random.str.zfill(20))
df
Out[9]: 
                 Random
0             000000086
1  00000000007639103627
2             000000096
3             000000032
4  00000000001469476501

Apply the same zfill function in a list comprehension, like this

>>> [str(item).zfill(6) for item in data]
['000001', '000010', '000313', '004000', '051234', '123456']

Alternatively, you can use the string's format method, with format specifiers, like this

>>> ["{:06d}".format(item) for item in data]
['000001', '000010', '000313', '004000', '051234', '123456']

If you are going to do the formatting more often, then you can store that in a variable, like this

>>> formatter = "{:06d}".format
>>> [formatter(item) for item in data]
['000001', '000010', '000313', '004000', '051234', '123456']

If you are using Python 2.x, then you can use map and the formatter function, like this

>>> map(formatter, data)
['000001', '000010', '000313', '004000', '051234', '123456']

If you are using Python 3.x, map returns an iterable map object. So, you need to explicitly create a list, like this

>>> list(map(formatter, data))
['000001', '000010', '000313', '004000', '051234', '123456']

Looking at Text | Text functions | ArcGIS Arcade | ArcGIS Developers

Text

Text(value, format?) -> Text

The format parameter is where you can specify the number of digits. For example:

var num = 12004;
return Text(num, '000000');

yields:

012004

If you want to make it dynamic, it is a bit cumbersome but doable:

var num = 12004;
var pad_len = 6;
var pad_char = 0;
return Text(num,  Concatenate(array(pad_len,pad_char)));

yields:

012004