First, to keep the cell reference fixed, use $A$1. As to getting autofill to increment the second half of the formula, I can’t get Excel to do that.
Unless someone else can, what you’ll need to do is use autofill to create the column of numbers you want to multiply $A$1 by, then in the top cell of the column next to that column, write a formula that multiplies $A$1 by the contents of the cell to the left of that column. Then you can copy that formula down.
Column C: Fill it with your number increment (say, 1 to 100). You can do this by entering 1 in cell C1, and then highlighting the cells you wish to fill with numbers from 1 to 100 and then going to Edit > Fill > Series. Include C1 in your highlight because the fill needs to know what number you want to start off with.
Column D: =CONCATENATE("$A$1*",C1) in cell D1. Copy this down to the bottom in parallel with column C. The result should look like this:
A B C D
[blank] [blank] 1 $A$1*1
[blank] [blank] 2 $A$1*2
[blank] [blank] 3 $A$1*3
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[blank] [blank] 100 $A$1*100
Now, copy the contents of column D over to column B by using Copy and then Paste Special > Values.
Highlight column B and bring up the Find/Replace dialog box. Enter “$A” in “Find” and “=$A” in replace.