My go-to tool for raster reprojection is gdalwarp, since gdal supports a myriad of formats including ASCII grid format.
For the task at hand (warping a *.asc dataset from EPSG:32629 to other CRS) I would open up a text console and run something like:
gdalwarp -s_srs epsg:32629 -t_srs epsg:4326 raster.asc warped.geotiff
ESRI is great software, but, yes, be prepared to shell out a pretty penny. ESRI is a lot more user friendly and, in my opinion, when it comes to pretty maps, ESRI is on top. That being said, there are a few open source solutions out there that can compare to ESRI software. Some are easier to learn than others and some have a bit a steep learning curve.