"I'm happy with a home tie," he said. "I don't really mind who we play but it's good that we're at home.
"I'd prefer not to play a team from our own division as well, so we've got to be pleased on both counts.
"I'd have been happy to play a Premier League team or a lower league club and we've got a lower league club so that's good.
"The draw has been kind to us so far in the FA Cup, Hastings at home and now Aldershot. But we've still got a job to do."
The tie pits Boro against a side who are currently 21st in League Two and who they have never played in any competition.
They reached this stage of the FA Cup by winning 2-1 at home to Hendon in round one, 3-2 away to Fleetwood in round two and 3-1 at home to Rotherham United in the third round on Saturday.
"Aldershot will be looking forward to the game," said Mowbray. "I think they're struggling a bit in League Two but we'll treat them with respect, as we would any side we're drawn against.
"The Cup's all about getting through to the next round and we've just got to win the game.
"(Aldershot manager) Dean Holdsworth has a bit of FA Cup history from his playing days.
"I haven't had a chance to look at them yet but I'll start to do that over the next week or so and we'll see what sort of football team we will come up against."
Boro's home game against Leeds United on Saturday January 26 will now have to be moved to allow the fourth round tie to take place and victory would mean the visit to Crystal Palace on Saturday February 16 would also be switched.
"The negative is having a game postponed against Leeds and if we win it will mean another fixture change and we'll play Crystal Palace in midweek, which is a very difficult journey to make," said Mowbray.
"Cardiff are out of the Cup now and they could keep winning games and stretch their lead at the top, leaving us with games in hand. But we have to be positive."
Although the sides have never met in any competition. Aldershot is the birthplace of one of Boro's most iconic managers, Bruce Rioch.
Draw in full…
Norwich City v Luton Town
Oldham Athletic v Mansfield Town or Liverpool
Macclesfield Town v Wigan Athletic or Bournemouth
Derby County v Blackburn Rovers
Hull City or Leyton Orient v Barnsley
BORO v Aldershot Town
Millwall v Aston Villa
Leeds United/Birmingham City v Tottenham Hotspur
Brighton v Swansea/Arsenal
Crystal Palace or Stoke City v Manchester City
Manchester United/West Ham United v Fulham/Blackpool
Southend United/Brentford v Chelsea
Reading v Sheffield United
Huddersfield Town v Leicester City
Queens Park Rangers/West Bromwich Albion v Sheffield Wednesday
Bolton/Sunderland v Cheltenham/Everton
Games will be played the weekend of Saturday January 26.